Linear Algebra Examples

$x^{2} + (1 - a) x - a = 0$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$x^{2} + 1 x - a x - a = 0$

Step 1.2

Multiply $x$ by $1$ .

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $- 1$ by $1$ .

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

Step 4.1.3

Multiply $- - a$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.3.2

Multiply $a$ by $1$ .

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

Step 4.1.4

Rewrite ${(1 - a)}^{2}$ as $(1 - a) (1 - a)$ .

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

Step 4.1.5

Expand $(1 - a) (1 - a)$ using the FOIL Method.

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Step 4.1.5.1

Apply the distributive property.

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

Step 4.1.5.2

Apply the distributive property.

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

Step 4.1.5.3

Apply the distributive property.

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

Step 4.1.6

Simplify and combine like terms.

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Step 4.1.6.1

Simplify each term.

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Step 4.1.6.1.1

Multiply $1$ by $1$ .

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

Step 4.1.6.1.2

Multiply $- a$ by $1$ .

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

Step 4.1.6.1.3

Multiply $- 1$ by $1$ .

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

Step 4.1.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.1.6.1.5

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

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Step 4.1.6.1.5.1

Move $a$ .

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

Step 4.1.6.1.5.2

Multiply $a$ by $a$ .

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

Step 4.1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 4.1.6.1.7

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

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

Step 4.1.6.2

Subtract $a$ from $- a$ .

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

Step 4.1.7

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

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Step 4.1.7.1

Multiply $- 4$ by $1$ .

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

Step 4.1.7.2

Multiply $- 4$ by $- 1$ .

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

Step 4.1.8

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

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

Step 4.1.9

Reorder terms.

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

Step 4.1.10

Factor using the perfect square rule.

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Step 4.1.10.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.10.2

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

$2 a = 2 \cdot a \cdot 1$

Step 4.1.10.3

Rewrite the polynomial.

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

Step 4.1.10.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

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

Step 4.1.11

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

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

Step 4.2

Multiply $2$ by $1$ .

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

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Multiply $- 1$ by $1$ .

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

Step 5.1.3

Multiply $- - a$ .

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.3.2

Multiply $a$ by $1$ .

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

Step 5.1.4

Rewrite ${(1 - a)}^{2}$ as $(1 - a) (1 - a)$ .

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

Step 5.1.5

Expand $(1 - a) (1 - a)$ using the FOIL Method.

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Step 5.1.5.1

Apply the distributive property.

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

Step 5.1.5.2

Apply the distributive property.

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

Step 5.1.5.3

Apply the distributive property.

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

Step 5.1.6

Simplify and combine like terms.

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Step 5.1.6.1

Simplify each term.

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Step 5.1.6.1.1

Multiply $1$ by $1$ .

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

Step 5.1.6.1.2

Multiply $- a$ by $1$ .

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

Step 5.1.6.1.3

Multiply $- 1$ by $1$ .

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

Step 5.1.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.1.6.1.5

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

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Step 5.1.6.1.5.1

Move $a$ .

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

Step 5.1.6.1.5.2

Multiply $a$ by $a$ .

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

Step 5.1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 5.1.6.1.7

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

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

Step 5.1.6.2

Subtract $a$ from $- a$ .

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

Step 5.1.7

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

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Step 5.1.7.1

Multiply $- 4$ by $1$ .

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

Step 5.1.7.2

Multiply $- 4$ by $- 1$ .

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

Step 5.1.8

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

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

Step 5.1.9

Reorder terms.

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

Step 5.1.10

Factor using the perfect square rule.

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Step 5.1.10.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.1.10.2

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

$2 a = 2 \cdot a \cdot 1$

Step 5.1.10.3

Rewrite the polynomial.

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

Step 5.1.10.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

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

Step 5.1.11

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

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

Change the $\pm$ to $+$ .

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Add $- 1$ and $1$ .

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

Step 5.4.2

Add $a + a$ and $0$ .

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

Step 5.4.3

Add $a$ and $a$ .

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

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Cancel the common factor.

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

Step 5.5.2

Divide $a$ by $1$ .

$x = a$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $- 1$ by $1$ .

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

Step 6.1.3

Multiply $- - a$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.3.2

Multiply $a$ by $1$ .

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

Step 6.1.4

Rewrite ${(1 - a)}^{2}$ as $(1 - a) (1 - a)$ .

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

Step 6.1.5

Expand $(1 - a) (1 - a)$ using the FOIL Method.

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Step 6.1.5.1

Apply the distributive property.

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

Step 6.1.5.2

Apply the distributive property.

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

Step 6.1.5.3

Apply the distributive property.

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

Step 6.1.6

Simplify and combine like terms.

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Step 6.1.6.1

Simplify each term.

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Step 6.1.6.1.1

Multiply $1$ by $1$ .

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

Step 6.1.6.1.2

Multiply $- a$ by $1$ .

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

Step 6.1.6.1.3

Multiply $- 1$ by $1$ .

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

Step 6.1.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 6.1.6.1.5

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

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Step 6.1.6.1.5.1

Move $a$ .

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

Step 6.1.6.1.5.2

Multiply $a$ by $a$ .

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

Step 6.1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 6.1.6.1.7

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

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

Step 6.1.6.2

Subtract $a$ from $- a$ .

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

Step 6.1.7

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

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Step 6.1.7.1

Multiply $- 4$ by $1$ .

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

Step 6.1.7.2

Multiply $- 4$ by $- 1$ .

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

Step 6.1.8

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

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

Step 6.1.9

Reorder terms.

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

Step 6.1.10

Factor using the perfect square rule.

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Step 6.1.10.1

Rewrite $1$ as $1^{2}$ .

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

Step 6.1.10.2

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

$2 a = 2 \cdot a \cdot 1$

Step 6.1.10.3

Rewrite the polynomial.

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

Step 6.1.10.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

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

Step 6.1.11

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

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

Change the $\pm$ to $-$ .

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

Step 6.4

Simplify the numerator.

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Step 6.4.1

Apply the distributive property.

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

Step 6.4.2

Multiply $- 1$ by $1$ .

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

Step 6.4.3

Subtract $1$ from $- 1$ .

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

Step 6.4.4

Subtract $a$ from $a$ .

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

Step 6.4.5

Subtract $2$ from $0$ .

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

Step 6.5

Divide $- 2$ by $2$ .

$x = - 1$

Step 7

The final answer is the combination of both solutions.

$x = a$

$x = - 1$

Step 8

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${a | a \in ℝ}$

Step 9