Finite Math Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - b, x - a, 1, 1$

Step 1.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.4

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.5

The factor for $x - b$ is $x - b$ itself.

$(x - b) = x - b$

$(x - b)$ occurs $1$ time.

Step 1.6

The factor for $x - a$ is $x - a$ itself.

$(x - a) = x - a$

$(x - a)$ occurs $1$ time.

Step 1.7

The LCM of $x - b, x - a$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x - b) (x - a)$

Step 2

Multiply each term in $\frac{a}{x - b} + \frac{b}{x - a} - 2 = 0$ by $(x - b) (x - a)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{a}{x - b} + \frac{b}{x - a} - 2 = 0$ by $(x - b) (x - a)$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $x - b$ .

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Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

Rewrite the expression.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Rewrite using the commutative property of multiplication.

$a x - a \cdot a + \frac{b}{x - a} ((x - b) (x - a)) - 2 ((x - b) (x - a)) = 0 ((x - b) (x - a))$

Step 2.2.1.4

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

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Step 2.2.1.4.1

Move $a$ .

$a x - (a \cdot a) + \frac{b}{x - a} ((x - b) (x - a)) - 2 ((x - b) (x - a)) = 0 ((x - b) (x - a))$

Step 2.2.1.4.2

Multiply $a$ by $a$ .

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

Step 2.2.1.5

Cancel the common factor of $x - a$ .

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Step 2.2.1.5.1

Factor $x - a$ out of $(x - b) (x - a)$ .

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

Step 2.2.1.5.2

Cancel the common factor.

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

Step 2.2.1.5.3

Rewrite the expression.

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

Step 2.2.1.6

Apply the distributive property.

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

Step 2.2.1.7

Rewrite using the commutative property of multiplication.

$a x - a^{2} + b x - b \cdot b - 2 ((x - b) (x - a)) = 0 ((x - b) (x - a))$

Step 2.2.1.8

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

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Step 2.2.1.8.1

Move $b$ .

$a x - a^{2} + b x - (b \cdot b) - 2 ((x - b) (x - a)) = 0 ((x - b) (x - a))$

Step 2.2.1.8.2

Multiply $b$ by $b$ .

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

Step 2.2.1.9

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

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Step 2.2.1.9.1

Apply the distributive property.

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

Step 2.2.1.9.2

Apply the distributive property.

$a x - a^{2} + b x - b^{2} - 2 (x \cdot x + x (- a) - b (x - a)) = 0 ((x - b) (x - a))$

Step 2.2.1.9.3

Apply the distributive property.

$a x - a^{2} + b x - b^{2} - 2 (x \cdot x + x (- a) - b x - b (- a)) = 0 ((x - b) (x - a))$

Step 2.2.1.10

Simplify each term.

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Step 2.2.1.10.1

Multiply $x$ by $x$ .

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

Step 2.2.1.10.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.10.3

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.10.4

Multiply $- 1$ by $- 1$ .

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

Step 2.2.1.10.5

Multiply $b$ by $1$ .

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

Step 2.2.1.11

Apply the distributive property.

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

Step 2.2.1.12

Simplify.

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Step 2.2.1.12.1

Multiply $- 1$ by $- 2$ .

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

Step 2.2.1.12.2

Multiply $- 1$ by $- 2$ .

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

Step 2.2.1.13

Remove parentheses.

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

Step 2.2.2

Add $a x$ and $2 x a$ .

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Step 2.2.2.1

Move $x$ .

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

Step 2.2.2.2

Add $a x$ and $2 a x$ .

$- a^{2} + b x - b^{2} - 2 x^{2} + 3 a x + 2 b x - 2 b a = 0 ((x - b) (x - a))$

Step 2.2.3

Add $b x$ and $2 b x$ .

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 ((x - b) (x - a))$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x (x - a) - b (x - a))$

Step 2.3.1.2

Apply the distributive property.

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x \cdot x + x (- a) - b (x - a))$

Step 2.3.1.3

Apply the distributive property.

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x \cdot x + x (- a) - b x - b (- a))$

Step 2.3.2

Simplify terms.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x$ by $x$ .

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x^{2} + x (- a) - b x - b (- a))$

Step 2.3.2.1.2

Rewrite using the commutative property of multiplication.

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x^{2} - x a - b x - b (- a))$

Step 2.3.2.1.3

Rewrite using the commutative property of multiplication.

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x^{2} - x a - b x - 1 \cdot - 1 b a)$

Step 2.3.2.1.4

Multiply $- 1$ by $- 1$ .

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x^{2} - x a - b x + 1 b a)$

Step 2.3.2.1.5

Multiply $b$ by $1$ .

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0 (x^{2} - x a - b x + b a)$

Step 2.3.2.2

Multiply $0$ by $x^{2} - x a - b x + b a$ .

$- a^{2} - b^{2} - 2 x^{2} + 3 a x + 3 b x - 2 b a = 0$

Step 3

Solve the equation.

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Step 3.1

Use the quadratic formula to find the solutions.

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

Step 3.2

Substitute the values $a = - 2$ , $b = 3 a + 3 b$ , and $c = - a^{2} - b^{2} - 2 b a$ into the quadratic formula and solve for $x$ .

$\frac{- (3 a + 3 b) \pm \sqrt{{(3 a + 3 b)}^{2} - 4 \cdot (- 2 \cdot (- a^{2} - b^{2} - 2 b a))}}{2 \cdot - 2}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Apply the distributive property.

$x = \frac{- (3 a) - (3 b) \pm \sqrt{{(3 a + 3 b)}^{2} - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.2

Multiply $3$ by $- 1$ .

$x = \frac{- 3 a - (3 b) \pm \sqrt{{(3 a + 3 b)}^{2} - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.3

Multiply $3$ by $- 1$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{{(3 a + 3 b)}^{2} - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.4

Rewrite ${(3 a + 3 b)}^{2}$ as $(3 a + 3 b) (3 a + 3 b)$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{(3 a + 3 b) (3 a + 3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.5

Expand $(3 a + 3 b) (3 a + 3 b)$ using the FOIL Method.

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Step 3.3.1.5.1

Apply the distributive property.

$x = \frac{- 3 a - 3 b \pm \sqrt{3 a (3 a + 3 b) + 3 b (3 a + 3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.5.2

Apply the distributive property.

$x = \frac{- 3 a - 3 b \pm \sqrt{3 a (3 a) + 3 a (3 b) + 3 b (3 a + 3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.5.3

Apply the distributive property.

$x = \frac{- 3 a - 3 b \pm \sqrt{3 a (3 a) + 3 a (3 b) + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6

Simplify and combine like terms.

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Step 3.3.1.6.1

Simplify each term.

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Step 3.3.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{- 3 a - 3 b \pm \sqrt{3 \cdot (3 a \cdot a) + 3 a (3 b) + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.2

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

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Step 3.3.1.6.1.2.1

Move $a$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{3 \cdot (3 (a \cdot a)) + 3 a (3 b) + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.2.2

Multiply $a$ by $a$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{3 \cdot (3 a^{2}) + 3 a (3 b) + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.3

Multiply $3$ by $3$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 3 a (3 b) + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.4

Rewrite using the commutative property of multiplication.

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 3 \cdot (3 a b) + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.5

Multiply $3$ by $3$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 3 b (3 a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.6

Rewrite using the commutative property of multiplication.

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 3 \cdot (3 b a) + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.7

Multiply $3$ by $3$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 9 b a + 3 b (3 b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.8

Rewrite using the commutative property of multiplication.

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 9 b a + 3 \cdot (3 b \cdot b) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.9

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

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Step 3.3.1.6.1.9.1

Move $b$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 9 b a + 3 \cdot (3 (b \cdot b)) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.9.2

Multiply $b$ by $b$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 9 b a + 3 \cdot (3 b^{2}) - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.1.10

Multiply $3$ by $3$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 9 b a + 9 b^{2} - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.2

Add $9 a b$ and $9 b a$ .

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Step 3.3.1.6.2.1

Move $b$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 9 a b + 9 a b + 9 b^{2} - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.6.2.2

Add $9 a b$ and $9 a b$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 18 a b + 9 b^{2} - 4 \cdot - 2 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.7

Multiply $- 4$ by $- 2$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 18 a b + 9 b^{2} + 8 \cdot (- a^{2} - b^{2} - 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.8

Apply the distributive property.

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 18 a b + 9 b^{2} + 8 (- a^{2}) + 8 (- b^{2}) + 8 (- 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.9

Simplify.

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Step 3.3.1.9.1

Multiply $- 1$ by $8$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 18 a b + 9 b^{2} - 8 a^{2} + 8 (- b^{2}) + 8 (- 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.9.2

Multiply $- 1$ by $8$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 18 a b + 9 b^{2} - 8 a^{2} - 8 b^{2} + 8 (- 2 b a)}}{2 \cdot - 2}$

Step 3.3.1.9.3

Multiply $- 2$ by $8$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{9 a^{2} + 18 a b + 9 b^{2} - 8 a^{2} - 8 b^{2} - 16 (b a)}}{2 \cdot - 2}$

Step 3.3.1.10

Subtract $8 a^{2}$ from $9 a^{2}$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{a^{2} + 18 a b + 9 b^{2} - 8 b^{2} - 16 b a}}{2 \cdot - 2}$

Step 3.3.1.11

Subtract $16 b a$ from $18 a b$ .

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Step 3.3.1.11.1

Move $b$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{a^{2} + 9 b^{2} - 8 b^{2} + 18 a b - 16 a b}}{2 \cdot - 2}$

Step 3.3.1.11.2

Subtract $16 a b$ from $18 a b$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{a^{2} + 9 b^{2} - 8 b^{2} + 2 a b}}{2 \cdot - 2}$

Step 3.3.1.12

Subtract $8 b^{2}$ from $9 b^{2}$ .

$x = \frac{- 3 a - 3 b \pm \sqrt{a^{2} + b^{2} + 2 a b}}{2 \cdot - 2}$

Step 3.3.1.13

Factor using the perfect square rule.

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Step 3.3.1.13.1

Rearrange terms.

$x = \frac{- 3 a - 3 b \pm \sqrt{a^{2} + 2 a b + b^{2}}}{2 \cdot - 2}$

Step 3.3.1.13.2

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

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

Step 3.3.1.13.3

Rewrite the polynomial.

$x = \frac{- 3 a - 3 b \pm \sqrt{a^{2} + 2 \cdot a \cdot b + b^{2}}}{2 \cdot - 2}$

Step 3.3.1.13.4

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

$x = \frac{- 3 a - 3 b \pm \sqrt{{(a + b)}^{2}}}{2 \cdot - 2}$

Step 3.3.1.14

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

$x = \frac{- 3 a - 3 b \pm (a + b)}{2 \cdot - 2}$

Step 3.3.2

Multiply $2$ by $- 2$ .

$x = \frac{- 3 a - 3 b \pm (a + b)}{- 4}$

Step 3.3.3

Simplify $\frac{- 3 a - 3 b \pm (a + b)}{- 4}$ .

$x = \frac{3 a + 3 b \pm (- (- a - b))}{4}$

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

Apply the distributive property.

$x = \frac{3 a + 3 b \pm (a + b)}{4}$

Step 3.3.4.2

Multiply $- - a$ .

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Step 3.3.4.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{3 a + 3 b \pm (1 a + b)}{4}$

Step 3.3.4.2.2

Multiply $a$ by $1$ .

$x = \frac{3 a + 3 b \pm (a + b)}{4}$

Step 3.3.4.3

Multiply $- - b$ .

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Step 3.3.4.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{3 a + 3 b \pm (a + 1 b)}{4}$

Step 3.3.4.3.2

Multiply $b$ by $1$ .

$x = \frac{3 a + 3 b \pm (a + b)}{4}$

Step 3.4

The final answer is the combination of both solutions.

$x = a + b$

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

$x = a + b$

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