Precalculus Examples

$\frac{x^{2} - b x + a x - a b}{x^{2} + b x - a x - a b} \div \frac{x^{2} + b x + a x + a b}{x^{2} - b x - a x + a b}$

Step 1

Set the denominator in $\frac{x^{2} - b x + a x - a b}{x^{2} + b x - a x - a b}$ equal to $0$ to find where the expression is undefined.

$x^{2} + b x - a x - a b = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2

Substitute the values $a = 1$ , $b = b - a$ , and $c = - a b$ into the quadratic formula and solve for $x$ .

$\frac{- (b - a) \pm \sqrt{{(b - a)}^{2} - 4 \cdot (1 \cdot (- a b))}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- - a$ .

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Step 2.3.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- b + 1 a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $a$ by $1$ .

$x = \frac{- b + a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.3

Rewrite ${(b - a)}^{2}$ as $(b - a) (b - a)$ .

$x = \frac{- b + a \pm \sqrt{(b - a) (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.4

Expand $(b - a) (b - a)$ using the FOIL Method.

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Step 2.3.1.4.1

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b (b - a) - a (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.4.2

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b \cdot b + b (- a) - a (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.4.3

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b \cdot b + b (- a) - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5

Simplify and combine like terms.

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Step 2.3.1.5.1

Simplify each term.

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Step 2.3.1.5.1.1

Multiply $b$ by $b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} + b (- a) - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.1.2

Rewrite using the commutative property of multiplication.

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.1.3

Rewrite using the commutative property of multiplication.

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 2.3.1.5.1.4.1

Move $a$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.1.4.2

Multiply $a$ by $a$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.1.5

Multiply $- 1$ by $- 1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b + 1 a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.1.6

Multiply $a^{2}$ by $1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.2

Subtract $a b$ from $- b a$ .

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Step 2.3.1.5.2.1

Move $b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - a b - a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.5.2.2

Subtract $a b$ from $- a b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.6

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.3.1.6.1

Multiply $- 4$ by $1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} - 4 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.3.1.6.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} + 4 a b}}{2 \cdot 1}$

Step 2.3.1.7

Add $- 2 a b$ and $4 a b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} + a^{2} + 2 a b}}{2 \cdot 1}$

Step 2.3.1.8

Factor using the perfect square rule.

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Step 2.3.1.8.1

Rearrange terms.

$x = \frac{- b + a \pm \sqrt{b^{2} + 2 a b + a^{2}}}{2 \cdot 1}$

Step 2.3.1.8.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 2.3.1.8.3

Rewrite the polynomial.

$x = \frac{- b + a \pm \sqrt{b^{2} + 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 2.3.1.8.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{- b + a \pm \sqrt{{(b + a)}^{2}}}{2 \cdot 1}$

Step 2.3.1.9

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- b + a \pm (b + a)}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{- b + a \pm (b + a)}{2}$

Step 2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- - a$ .

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Step 2.4.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- b + 1 a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $a$ by $1$ .

$x = \frac{- b + a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.3

Rewrite ${(b - a)}^{2}$ as $(b - a) (b - a)$ .

$x = \frac{- b + a \pm \sqrt{(b - a) (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.4

Expand $(b - a) (b - a)$ using the FOIL Method.

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Step 2.4.1.4.1

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b (b - a) - a (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.4.2

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b \cdot b + b (- a) - a (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.4.3

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b \cdot b + b (- a) - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5

Simplify and combine like terms.

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Step 2.4.1.5.1

Simplify each term.

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Step 2.4.1.5.1.1

Multiply $b$ by $b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} + b (- a) - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.1.2

Rewrite using the commutative property of multiplication.

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.1.3

Rewrite using the commutative property of multiplication.

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 2.4.1.5.1.4.1

Move $a$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.1.4.2

Multiply $a$ by $a$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.1.5

Multiply $- 1$ by $- 1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b + 1 a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.1.6

Multiply $a^{2}$ by $1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.2

Subtract $a b$ from $- b a$ .

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Step 2.4.1.5.2.1

Move $b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - a b - a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.5.2.2

Subtract $a b$ from $- a b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.6

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.4.1.6.1

Multiply $- 4$ by $1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} - 4 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.4.1.6.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} + 4 a b}}{2 \cdot 1}$

Step 2.4.1.7

Add $- 2 a b$ and $4 a b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} + a^{2} + 2 a b}}{2 \cdot 1}$

Step 2.4.1.8

Factor using the perfect square rule.

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Step 2.4.1.8.1

Rearrange terms.

$x = \frac{- b + a \pm \sqrt{b^{2} + 2 a b + a^{2}}}{2 \cdot 1}$

Step 2.4.1.8.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 2.4.1.8.3

Rewrite the polynomial.

$x = \frac{- b + a \pm \sqrt{b^{2} + 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 2.4.1.8.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{- b + a \pm \sqrt{{(b + a)}^{2}}}{2 \cdot 1}$

Step 2.4.1.9

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- b + a \pm (b + a)}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{- b + a \pm (b + a)}{2}$

Step 2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- b + a + b + a}{2}$

Step 2.4.4

Simplify the numerator.

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Step 2.4.4.1

Add $- b$ and $b$ .

$x = \frac{a + 0 + a}{2}$

Step 2.4.4.2

Add $a$ and $0$ .

$x = \frac{a + a}{2}$

Step 2.4.4.3

Add $a$ and $a$ .

$x = \frac{2 a}{2}$

Step 2.4.5

Cancel the common factor of $2$ .

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Step 2.4.5.1

Cancel the common factor.

$x = \frac{2 a}{2}$

Step 2.4.5.2

Divide $a$ by $1$ .

$x = a$

Step 2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- - a$ .

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Step 2.5.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- b + 1 a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $a$ by $1$ .

$x = \frac{- b + a \pm \sqrt{{(b - a)}^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.3

Rewrite ${(b - a)}^{2}$ as $(b - a) (b - a)$ .

$x = \frac{- b + a \pm \sqrt{(b - a) (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.4

Expand $(b - a) (b - a)$ using the FOIL Method.

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Step 2.5.1.4.1

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b (b - a) - a (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.4.2

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b \cdot b + b (- a) - a (b - a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.4.3

Apply the distributive property.

$x = \frac{- b + a \pm \sqrt{b \cdot b + b (- a) - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5

Simplify and combine like terms.

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Step 2.5.1.5.1

Simplify each term.

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Step 2.5.1.5.1.1

Multiply $b$ by $b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} + b (- a) - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.1.2

Rewrite using the commutative property of multiplication.

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - a (- a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.1.3

Rewrite using the commutative property of multiplication.

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 2.5.1.5.1.4.1

Move $a$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.1.4.2

Multiply $a$ by $a$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.1.5

Multiply $- 1$ by $- 1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b + 1 a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.1.6

Multiply $a^{2}$ by $1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - b a - a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.2

Subtract $a b$ from $- b a$ .

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Step 2.5.1.5.2.1

Move $b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - a b - a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.5.2.2

Subtract $a b$ from $- a b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} - 4 \cdot 1 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.6

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.5.1.6.1

Multiply $- 4$ by $1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} - 4 \cdot (- 1 a b)}}{2 \cdot 1}$

Step 2.5.1.6.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- b + a \pm \sqrt{b^{2} - 2 a b + a^{2} + 4 a b}}{2 \cdot 1}$

Step 2.5.1.7

Add $- 2 a b$ and $4 a b$ .

$x = \frac{- b + a \pm \sqrt{b^{2} + a^{2} + 2 a b}}{2 \cdot 1}$

Step 2.5.1.8

Factor using the perfect square rule.

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Step 2.5.1.8.1

Rearrange terms.

$x = \frac{- b + a \pm \sqrt{b^{2} + 2 a b + a^{2}}}{2 \cdot 1}$

Step 2.5.1.8.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 2.5.1.8.3

Rewrite the polynomial.

$x = \frac{- b + a \pm \sqrt{b^{2} + 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 2.5.1.8.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{- b + a \pm \sqrt{{(b + a)}^{2}}}{2 \cdot 1}$

Step 2.5.1.9

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- b + a \pm (b + a)}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- b + a \pm (b + a)}{2}$

Step 2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- b + a - (b + a)}{2}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Apply the distributive property.

$x = \frac{- b + a - b - a}{2}$

Step 2.5.4.2

Subtract $b$ from $- b$ .

$x = \frac{a - 2 b - a}{2}$

Step 2.5.4.3

Subtract $a$ from $a$ .

$x = \frac{- 2 b + 0}{2}$

Step 2.5.4.4

Add $- 2 b$ and $0$ .

$x = \frac{- 2 b}{2}$

Step 2.5.5

Cancel the common factor of $- 2$ and $2$ .

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Step 2.5.5.1

Factor $2$ out of $- 2 b$ .

$x = \frac{2 (- b)}{2}$

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (- b)}{2 (1)}$

Step 2.5.5.2.2

Cancel the common factor.

$x = \frac{2 (- b)}{2 \cdot 1}$

Step 2.5.5.2.3

Rewrite the expression.

$x = \frac{- b}{1}$

Step 2.5.5.2.4

Divide $- b$ by $1$ .

$x = - b$

Step 2.6

The final answer is the combination of both solutions.

$x = a$

$x = - b$

$x = a$

$x = - b$

Step 3

Set the denominator in $\frac{x^{2} + b x + a x + a b}{x^{2} - b x - a x + a b}$ equal to $0$ to find where the expression is undefined.

$x^{2} - b x - a x + a b = 0$

Step 4

Solve for $x$ .

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Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2

Substitute the values $a = 1$ , $b = - b - a$ , and $c = a b$ into the quadratic formula and solve for $x$ .

$\frac{- (- b - a) \pm \sqrt{{(- b - a)}^{2} - 4 \cdot (1 \cdot (a b))}}{2 \cdot 1}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- - b$ .

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Step 4.3.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1 b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.2.2

Multiply $b$ by $1$ .

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.3

Multiply $- - a$ .

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Step 4.3.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{b + 1 a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.3.2

Multiply $a$ by $1$ .

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite ${(- b - a)}^{2}$ as $(- b - a) (- b - a)$ .

$x = \frac{b + a \pm \sqrt{(- b - a) (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.5

Expand $(- b - a) (- b - a)$ using the FOIL Method.

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Step 4.3.1.5.1

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b - a) - a (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.5.2

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b) - b (- a) - a (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.5.3

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6

Simplify and combine like terms.

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Step 4.3.1.6.1

Simplify each term.

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Step 4.3.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 b \cdot b) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.2

Multiply $b$ by $b$ by adding the exponents.

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Step 4.3.1.6.1.2.1

Move $b$ .

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 (b \cdot b)) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.2.2

Multiply $b$ by $b$ .

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 b^{2}) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{1 b^{2} - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.4

Multiply $b^{2}$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.5

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} - 1 \cdot (- 1 b a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 1 b a - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.7

Multiply $b$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.8

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} + b a - 1 \cdot (- 1 a b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.9

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + 1 a b - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.10

Multiply $a$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.11

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.12

Multiply $a$ by $a$ by adding the exponents.

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Step 4.3.1.6.1.12.1

Move $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.12.2

Multiply $a$ by $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.13

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b + 1 a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.1.14

Multiply $a^{2}$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.2

Add $b a$ and $a b$ .

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Step 4.3.1.6.2.1

Reorder $b$ and $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + a b + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.6.2.2

Add $a b$ and $a b$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.7

Multiply $- 4$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot (a b)}}{2 \cdot 1}$

Step 4.3.1.8

Subtract $4 a b$ from $2 a b$ .

$x = \frac{b + a \pm \sqrt{b^{2} + a^{2} - 2 a b}}{2 \cdot 1}$

Step 4.3.1.9

Factor using the perfect square rule.

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Step 4.3.1.9.1

Rearrange terms.

$x = \frac{b + a \pm \sqrt{b^{2} - 2 a b + a^{2}}}{2 \cdot 1}$

Step 4.3.1.9.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 4.3.1.9.3

Rewrite the polynomial.

$x = \frac{b + a \pm \sqrt{b^{2} - 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 4.3.1.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{b + a \pm \sqrt{{(b - a)}^{2}}}{2 \cdot 1}$

Step 4.3.1.10

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{b + a \pm (b - a)}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{b + a \pm (b - a)}{2}$

Step 4.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- - b$ .

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Step 4.4.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1 b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $b$ by $1$ .

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.3

Multiply $- - a$ .

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Step 4.4.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{b + 1 a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.3.2

Multiply $a$ by $1$ .

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite ${(- b - a)}^{2}$ as $(- b - a) (- b - a)$ .

$x = \frac{b + a \pm \sqrt{(- b - a) (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.5

Expand $(- b - a) (- b - a)$ using the FOIL Method.

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Step 4.4.1.5.1

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b - a) - a (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.5.2

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b) - b (- a) - a (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.5.3

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6

Simplify and combine like terms.

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Step 4.4.1.6.1

Simplify each term.

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Step 4.4.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 b \cdot b) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.2

Multiply $b$ by $b$ by adding the exponents.

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Step 4.4.1.6.1.2.1

Move $b$ .

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 (b \cdot b)) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.2.2

Multiply $b$ by $b$ .

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 b^{2}) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{1 b^{2} - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.4

Multiply $b^{2}$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.5

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} - 1 \cdot (- 1 b a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 1 b a - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.7

Multiply $b$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.8

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} + b a - 1 \cdot (- 1 a b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.9

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + 1 a b - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.10

Multiply $a$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.11

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.12

Multiply $a$ by $a$ by adding the exponents.

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Step 4.4.1.6.1.12.1

Move $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.12.2

Multiply $a$ by $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.13

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b + 1 a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.1.14

Multiply $a^{2}$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.2

Add $b a$ and $a b$ .

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Step 4.4.1.6.2.1

Reorder $b$ and $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + a b + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.6.2.2

Add $a b$ and $a b$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.7

Multiply $- 4$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot (a b)}}{2 \cdot 1}$

Step 4.4.1.8

Subtract $4 a b$ from $2 a b$ .

$x = \frac{b + a \pm \sqrt{b^{2} + a^{2} - 2 a b}}{2 \cdot 1}$

Step 4.4.1.9

Factor using the perfect square rule.

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Step 4.4.1.9.1

Rearrange terms.

$x = \frac{b + a \pm \sqrt{b^{2} - 2 a b + a^{2}}}{2 \cdot 1}$

Step 4.4.1.9.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 4.4.1.9.3

Rewrite the polynomial.

$x = \frac{b + a \pm \sqrt{b^{2} - 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 4.4.1.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{b + a \pm \sqrt{{(b - a)}^{2}}}{2 \cdot 1}$

Step 4.4.1.10

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{b + a \pm (b - a)}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{b + a \pm (b - a)}{2}$

Step 4.4.3

Change the $\pm$ to $+$ .

$x = \frac{b + a + b - a}{2}$

Step 4.4.4

Simplify the numerator.

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Step 4.4.4.1

Add $b$ and $b$ .

$x = \frac{a + 2 b - a}{2}$

Step 4.4.4.2

Subtract $a$ from $a$ .

$x = \frac{2 b + 0}{2}$

Step 4.4.4.3

Add $2 b$ and $0$ .

$x = \frac{2 b}{2}$

Step 4.4.5

Cancel the common factor of $2$ .

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Step 4.4.5.1

Cancel the common factor.

$x = \frac{2 b}{2}$

Step 4.4.5.2

Divide $b$ by $1$ .

$x = b$

Step 4.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- - b$ .

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Step 4.5.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1 b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $b$ by $1$ .

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.3

Multiply $- - a$ .

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Step 4.5.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{b + 1 a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.3.2

Multiply $a$ by $1$ .

$x = \frac{b + a \pm \sqrt{{(- b - a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite ${(- b - a)}^{2}$ as $(- b - a) (- b - a)$ .

$x = \frac{b + a \pm \sqrt{(- b - a) (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.5

Expand $(- b - a) (- b - a)$ using the FOIL Method.

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Step 4.5.1.5.1

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b - a) - a (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.5.2

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b) - b (- a) - a (- b - a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.5.3

Apply the distributive property.

$x = \frac{b + a \pm \sqrt{- b (- b) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6

Simplify and combine like terms.

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Step 4.5.1.6.1

Simplify each term.

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Step 4.5.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 b \cdot b) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.2

Multiply $b$ by $b$ by adding the exponents.

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Step 4.5.1.6.1.2.1

Move $b$ .

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 (b \cdot b)) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.2.2

Multiply $b$ by $b$ .

$x = \frac{b + a \pm \sqrt{- 1 \cdot (- 1 b^{2}) - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{1 b^{2} - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.4

Multiply $b^{2}$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} - b (- a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.5

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} - 1 \cdot (- 1 b a) - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 1 b a - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.7

Multiply $b$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a - a (- b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.8

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} + b a - 1 \cdot (- 1 a b) - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.9

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + 1 a b - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.10

Multiply $a$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - a (- a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.11

Rewrite using the commutative property of multiplication.

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.12

Multiply $a$ by $a$ by adding the exponents.

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Step 4.5.1.6.1.12.1

Move $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.12.2

Multiply $a$ by $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.13

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b + 1 a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.1.14

Multiply $a^{2}$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + b a + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.2

Add $b a$ and $a b$ .

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Step 4.5.1.6.2.1

Reorder $b$ and $a$ .

$x = \frac{b + a \pm \sqrt{b^{2} + a b + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.6.2.2

Add $a b$ and $a b$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.7

Multiply $- 4$ by $1$ .

$x = \frac{b + a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot (a b)}}{2 \cdot 1}$

Step 4.5.1.8

Subtract $4 a b$ from $2 a b$ .

$x = \frac{b + a \pm \sqrt{b^{2} + a^{2} - 2 a b}}{2 \cdot 1}$

Step 4.5.1.9

Factor using the perfect square rule.

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Step 4.5.1.9.1

Rearrange terms.

$x = \frac{b + a \pm \sqrt{b^{2} - 2 a b + a^{2}}}{2 \cdot 1}$

Step 4.5.1.9.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 4.5.1.9.3

Rewrite the polynomial.

$x = \frac{b + a \pm \sqrt{b^{2} - 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 4.5.1.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{b + a \pm \sqrt{{(b - a)}^{2}}}{2 \cdot 1}$

Step 4.5.1.10

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{b + a \pm (b - a)}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{b + a \pm (b - a)}{2}$

Step 4.5.3

Change the $\pm$ to $-$ .

$x = \frac{b + a - (b - a)}{2}$

Step 4.5.4

Simplify the numerator.

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Step 4.5.4.1

Apply the distributive property.

$x = \frac{b + a - b + a}{2}$

Step 4.5.4.2

Multiply $- - a$ .

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Step 4.5.4.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{b + a - b + 1 a}{2}$

Step 4.5.4.2.2

Multiply $a$ by $1$ .

$x = \frac{b + a - b + a}{2}$

Step 4.5.4.3

Subtract $b$ from $b$ .

$x = \frac{a + 0 + a}{2}$

Step 4.5.4.4

Add $a$ and $0$ .

$x = \frac{a + a}{2}$

Step 4.5.4.5

Add $a$ and $a$ .

$x = \frac{2 a}{2}$

Step 4.5.5

Cancel the common factor of $2$ .

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Step 4.5.5.1

Cancel the common factor.

$x = \frac{2 a}{2}$

Step 4.5.5.2

Divide $a$ by $1$ .

$x = a$

Step 4.6

The final answer is the combination of both solutions.

$x = b$

$x = a$

$x = b$

$x = a$

Step 5

Set the denominator in $\frac{x^{2} - b x + a x - a b}{x^{2} + b x - a x - a b} \div \frac{x^{2} + b x + a x + a b}{x^{2} - b x - a x + a b}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} + b x + a x + a b}{x^{2} - b x - a x + a b} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Set the numerator equal to zero.

$x^{2} + b x + a x + a b = 0$

Step 6.2

Solve the equation for $x$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.2

Substitute the values $a = 1$ , $b = b + a$ , and $c = a b$ into the quadratic formula and solve for $x$ .

$\frac{- (b + a) \pm \sqrt{{(b + a)}^{2} - 4 \cdot (1 \cdot (a b))}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{{(b + a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.2

Rewrite ${(b + a)}^{2}$ as $(b + a) (b + a)$ .

$x = \frac{- b - a \pm \sqrt{(b + a) (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.3

Expand $(b + a) (b + a)$ using the FOIL Method.

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Step 6.2.3.1.3.1

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b (b + a) + a (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.3.2

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b \cdot b + b a + a (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.3.3

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b \cdot b + b a + a b + a \cdot a - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.4

Simplify and combine like terms.

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Step 6.2.3.1.4.1

Simplify each term.

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Step 6.2.3.1.4.1.1

Multiply $b$ by $b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + b a + a b + a \cdot a - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.4.1.2

Multiply $a$ by $a$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + b a + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.4.2

Add $b a$ and $a b$ .

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Step 6.2.3.1.4.2.1

Reorder $b$ and $a$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + a b + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.4.2.2

Add $a b$ and $a b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.5

Multiply $- 4$ by $1$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.3.1.6

Subtract $4 a b$ from $2 a b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + a^{2} - 2 a b}}{2 \cdot 1}$

Step 6.2.3.1.7

Factor using the perfect square rule.

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Step 6.2.3.1.7.1

Rearrange terms.

$x = \frac{- b - a \pm \sqrt{b^{2} - 2 a b + a^{2}}}{2 \cdot 1}$

Step 6.2.3.1.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 6.2.3.1.7.3

Rewrite the polynomial.

$x = \frac{- b - a \pm \sqrt{b^{2} - 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 6.2.3.1.7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{- b - a \pm \sqrt{{(b - a)}^{2}}}{2 \cdot 1}$

Step 6.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- b - a \pm (b - a)}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- b - a \pm (b - a)}{2}$

Step 6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{{(b + a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.2

Rewrite ${(b + a)}^{2}$ as $(b + a) (b + a)$ .

$x = \frac{- b - a \pm \sqrt{(b + a) (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.3

Expand $(b + a) (b + a)$ using the FOIL Method.

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Step 6.2.4.1.3.1

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b (b + a) + a (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.3.2

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b \cdot b + b a + a (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.3.3

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b \cdot b + b a + a b + a \cdot a - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.4

Simplify and combine like terms.

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Step 6.2.4.1.4.1

Simplify each term.

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Step 6.2.4.1.4.1.1

Multiply $b$ by $b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + b a + a b + a \cdot a - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.4.1.2

Multiply $a$ by $a$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + b a + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.4.2

Add $b a$ and $a b$ .

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Step 6.2.4.1.4.2.1

Reorder $b$ and $a$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + a b + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.4.2.2

Add $a b$ and $a b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.5

Multiply $- 4$ by $1$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.4.1.6

Subtract $4 a b$ from $2 a b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + a^{2} - 2 a b}}{2 \cdot 1}$

Step 6.2.4.1.7

Factor using the perfect square rule.

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Step 6.2.4.1.7.1

Rearrange terms.

$x = \frac{- b - a \pm \sqrt{b^{2} - 2 a b + a^{2}}}{2 \cdot 1}$

Step 6.2.4.1.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 6.2.4.1.7.3

Rewrite the polynomial.

$x = \frac{- b - a \pm \sqrt{b^{2} - 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 6.2.4.1.7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{- b - a \pm \sqrt{{(b - a)}^{2}}}{2 \cdot 1}$

Step 6.2.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- b - a \pm (b - a)}{2 \cdot 1}$

Step 6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- b - a \pm (b - a)}{2}$

Step 6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- b - a + b - a}{2}$

Step 6.2.4.4

Simplify the numerator.

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Step 6.2.4.4.1

Add $- b$ and $b$ .

$x = \frac{- a + 0 - a}{2}$

Step 6.2.4.4.2

Add $- a$ and $0$ .

$x = \frac{- a - a}{2}$

Step 6.2.4.4.3

Subtract $a$ from $- a$ .

$x = \frac{- 2 a}{2}$

Step 6.2.4.5

Cancel the common factor of $- 2$ and $2$ .

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Step 6.2.4.5.1

Factor $2$ out of $- 2 a$ .

$x = \frac{2 (- a)}{2}$

Step 6.2.4.5.2

Cancel the common factors.

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Step 6.2.4.5.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (- a)}{2 (1)}$

Step 6.2.4.5.2.2

Cancel the common factor.

$x = \frac{2 (- a)}{2 \cdot 1}$

Step 6.2.4.5.2.3

Rewrite the expression.

$x = \frac{- a}{1}$

Step 6.2.4.5.2.4

Divide $- a$ by $1$ .

$x = - a$

Step 6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{{(b + a)}^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.2

Rewrite ${(b + a)}^{2}$ as $(b + a) (b + a)$ .

$x = \frac{- b - a \pm \sqrt{(b + a) (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.3

Expand $(b + a) (b + a)$ using the FOIL Method.

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Step 6.2.5.1.3.1

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b (b + a) + a (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.3.2

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b \cdot b + b a + a (b + a) - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.3.3

Apply the distributive property.

$x = \frac{- b - a \pm \sqrt{b \cdot b + b a + a b + a \cdot a - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.4

Simplify and combine like terms.

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Step 6.2.5.1.4.1

Simplify each term.

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Step 6.2.5.1.4.1.1

Multiply $b$ by $b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + b a + a b + a \cdot a - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.4.1.2

Multiply $a$ by $a$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + b a + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.4.2

Add $b a$ and $a b$ .

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Step 6.2.5.1.4.2.1

Reorder $b$ and $a$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + a b + a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.4.2.2

Add $a b$ and $a b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot 1 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.5

Multiply $- 4$ by $1$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + 2 a b + a^{2} - 4 \cdot (a b)}}{2 \cdot 1}$

Step 6.2.5.1.6

Subtract $4 a b$ from $2 a b$ .

$x = \frac{- b - a \pm \sqrt{b^{2} + a^{2} - 2 a b}}{2 \cdot 1}$

Step 6.2.5.1.7

Factor using the perfect square rule.

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Step 6.2.5.1.7.1

Rearrange terms.

$x = \frac{- b - a \pm \sqrt{b^{2} - 2 a b + a^{2}}}{2 \cdot 1}$

Step 6.2.5.1.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Step 6.2.5.1.7.3

Rewrite the polynomial.

$x = \frac{- b - a \pm \sqrt{b^{2} - 2 \cdot b \cdot a + a^{2}}}{2 \cdot 1}$

Step 6.2.5.1.7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{- b - a \pm \sqrt{{(b - a)}^{2}}}{2 \cdot 1}$

Step 6.2.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- b - a \pm (b - a)}{2 \cdot 1}$

Step 6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- b - a \pm (b - a)}{2}$

Step 6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- b - a - (b - a)}{2}$

Step 6.2.5.4

Simplify the numerator.

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Step 6.2.5.4.1

Apply the distributive property.

$x = \frac{- b - a - b + a}{2}$

Step 6.2.5.4.2

Multiply $- - a$ .

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Step 6.2.5.4.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- b - a - b + 1 a}{2}$

Step 6.2.5.4.2.2

Multiply $a$ by $1$ .

$x = \frac{- b - a - b + a}{2}$

Step 6.2.5.4.3

Subtract $b$ from $- b$ .

$x = \frac{- a - 2 b + a}{2}$

Step 6.2.5.4.4

Add $- a$ and $a$ .

$x = \frac{- 2 b + 0}{2}$

Step 6.2.5.4.5

Add $- 2 b$ and $0$ .

$x = \frac{- 2 b}{2}$

Step 6.2.5.5

Cancel the common factor of $- 2$ and $2$ .

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Step 6.2.5.5.1

Factor $2$ out of $- 2 b$ .

$x = \frac{2 (- b)}{2}$

Step 6.2.5.5.2

Cancel the common factors.

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Step 6.2.5.5.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (- b)}{2 (1)}$

Step 6.2.5.5.2.2

Cancel the common factor.

$x = \frac{2 (- b)}{2 \cdot 1}$

Step 6.2.5.5.2.3

Rewrite the expression.

$x = \frac{- b}{1}$

Step 6.2.5.5.2.4

Divide $- b$ by $1$ .

$x = - b$

Step 6.2.6

The final answer is the combination of both solutions.

$x = - a$

$x = - b$

$x = - a$

$x = - b$

$x = - a$

$x = - b$

Step 7

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$