Algebra Examples

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

Step 1

Multiply both sides by $x - 1$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{a (1 + \sqrt{x})}{x - 1} (x - 1)$ .

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Step 2.1.1.1

Cancel the common factor of $x - 1$ .

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Step 2.1.1.1.1

Cancel the common factor.

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

Step 2.1.1.1.2

Rewrite the expression.

$a (1 + \sqrt{x}) = b (x - 1)$

Step 2.1.1.2

Apply the distributive property.

$a \cdot 1 + a \sqrt{x} = b (x - 1)$

Step 2.1.1.3

Multiply $a$ by $1$ .

$a + a \sqrt{x} = b (x - 1)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $b (x - 1)$ .

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Step 2.2.1.1

Simplify by multiplying through.

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Step 2.2.1.1.1

Apply the distributive property.

$a + a \sqrt{x} = b x + b \cdot - 1$

Step 2.2.1.1.2

Move $- 1$ to the left of $b$ .

$a + a \sqrt{x} = b x - 1 \cdot b$

Step 2.2.1.2

Rewrite $- 1 b$ as $- b$ .

$a + a \sqrt{x} = b x - b$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $a$ from both sides of the equation.

$a \sqrt{x} = b x - b - a$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${(a x^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Apply the product rule to $a x^{\frac{1}{2}}$ .

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

Step 3.3.2.1.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3.2.1.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.2.1

Cancel the common factor.

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

Step 3.3.2.1.2.2.2

Rewrite the expression.

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

Step 3.3.2.1.3

Simplify.

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify ${(b x - b - a)}^{2}$ .

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Step 3.3.3.1.1

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

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

Step 3.3.3.1.2

Expand $(b x - b - a) (b x - b - a)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.3.3.1.3

Simplify terms.

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Step 3.3.3.1.3.1

Simplify each term.

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Step 3.3.3.1.3.1.1

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

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Step 3.3.3.1.3.1.1.1

Move $b$ .

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

Step 3.3.3.1.3.1.1.2

Multiply $b$ by $b$ .

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

Step 3.3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.3.1.3.1.2.1

Move $x$ .

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

Step 3.3.3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.3.1.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.3.1.4

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

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Step 3.3.3.1.3.1.4.1

Move $b$ .

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

Step 3.3.3.1.3.1.4.2

Multiply $b$ by $b$ .

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

Step 3.3.3.1.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.3.1.6

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

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Step 3.3.3.1.3.1.6.1

Move $b$ .

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

Step 3.3.3.1.3.1.6.2

Multiply $b$ by $b$ .

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

Step 3.3.3.1.3.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.3.1.8

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

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Step 3.3.3.1.3.1.8.1

Move $b$ .

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

Step 3.3.3.1.3.1.8.2

Multiply $b$ by $b$ .

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

Step 3.3.3.1.3.1.9

Multiply $- 1$ by $- 1$ .

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

Step 3.3.3.1.3.1.10

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

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

Step 3.3.3.1.3.1.11

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.3.1.12

Multiply $- 1$ by $- 1$ .

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

Step 3.3.3.1.3.1.13

Multiply $b$ by $1$ .

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

Step 3.3.3.1.3.1.14

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.3.1.15

Multiply $- 1$ by $- 1$ .

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

Step 3.3.3.1.3.1.16

Multiply $a$ by $1$ .

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

Step 3.3.3.1.3.1.17

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.3.1.18

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

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Step 3.3.3.1.3.1.18.1

Move $a$ .

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

Step 3.3.3.1.3.1.18.2

Multiply $a$ by $a$ .

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

Step 3.3.3.1.3.1.19

Multiply $- 1$ by $- 1$ .

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

Step 3.3.3.1.3.1.20

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

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

Step 3.3.3.1.3.2

Subtract $b^{2} x$ from $- b^{2} x$ .

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

Step 3.3.3.1.4

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

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Step 3.3.3.1.4.1

Move $b$ .

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

Step 3.3.3.1.4.2

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

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

Step 3.3.3.1.5

Add $b a$ and $a b$ .

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Step 3.3.3.1.5.1

Reorder $b$ and $a$ .

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

Step 3.3.3.1.5.2

Add $a b$ and $a b$ .

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

Step 3.4

Solve for $x$ .

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Step 3.4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3.4.2

Subtract $a^{2} x$ from both sides of the equation.

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

Step 3.4.3

Use the quadratic formula to find the solutions.

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

Step 3.4.4

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

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

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Apply the distributive property.

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

Step 3.4.5.2

Simplify.

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Step 3.4.5.2.1

Multiply $- 2$ by $- 1$ .

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

Step 3.4.5.2.2

Multiply $- 2$ by $- 1$ .

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

Step 3.4.5.2.3

Multiply $- - a^{2}$ .

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Step 3.4.5.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.5.2.3.2

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

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

Step 3.4.5.3

Rewrite $4 \cdot b^{2} \cdot (b^{2} + 2 a b + a^{2})$ as ${(2 b (b + a))}^{2}$ .

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

Step 3.4.5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 2 b^{2} - 2 a b - a^{2}$ and $b = 2 b (b + a)$ .

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

Step 3.4.5.5

Simplify.

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Step 3.4.5.5.1

Apply the distributive property.

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

Step 3.4.5.5.2

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

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Step 3.4.5.5.2.1

Move $b$ .

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

Step 3.4.5.5.2.2

Multiply $b$ by $b$ .

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

Step 3.4.5.5.3

Add $- 2 b^{2}$ and $2 b^{2}$ .

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

Step 3.4.5.5.4

Add $- 2 a b$ and $0$ .

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

Step 3.4.5.5.5

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

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Step 3.4.5.5.5.1

Move $b$ .

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

Step 3.4.5.5.5.2

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

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

Step 3.4.5.5.6

Add $- a^{2}$ and $0$ .

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

Step 3.4.5.5.7

Multiply $2$ by $- 1$ .

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

Step 3.4.5.6

Simplify each term.

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Step 3.4.5.6.1

Apply the distributive property.

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

Step 3.4.5.6.2

Multiply $b$ by $b$ .

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

Step 3.4.5.6.3

Apply the distributive property.

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

Step 3.4.5.7

Subtract $2 b^{2}$ from $- 2 b^{2}$ .

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

Step 3.4.5.8

Subtract $2 b a$ from $- 2 a b$ .

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Step 3.4.5.8.1

Move $b$ .

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

Step 3.4.5.8.2

Subtract $2 a b$ from $- 2 a b$ .

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

Step 3.4.5.9

Factor by grouping.

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Step 3.4.5.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = - 4$ .

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Step 3.4.5.9.1.1

Reorder terms.

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

Step 3.4.5.9.1.2

Reorder $- 4 b^{2}$ and $- 4 a b$ .

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

Step 3.4.5.9.1.3

Factor $- 4$ out of $- 4 a b$ .

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

Step 3.4.5.9.1.4

Rewrite $- 4$ as $- 2$ plus $- 2$

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

Step 3.4.5.9.1.5

Apply the distributive property.

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

Step 3.4.5.9.1.6

Move parentheses.

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

Step 3.4.5.9.2

Factor out the greatest common factor from each group.

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Step 3.4.5.9.2.1

Group the first two terms and the last two terms.

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

Step 3.4.5.9.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.4.5.9.3

Factor the polynomial by factoring out the greatest common factor, $- a - 2 b$ .

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

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

Step 3.4.5.10

Combine exponents.

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Step 3.4.5.10.1

Factor $- 1$ out of $- a$ .

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

Step 3.4.5.10.2

Factor $- 1$ out of $- 2 b$ .

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

Step 3.4.5.10.3

Factor $- 1$ out of $- (a) - (2 b)$ .

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

Step 3.4.5.10.4

Rewrite $- (a + 2 b)$ as $- 1 (a + 2 b)$ .

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

Step 3.4.5.10.5

Raise $a + 2 b$ to the power of $1$ .

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

Step 3.4.5.10.6

Raise $a + 2 b$ to the power of $1$ .

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

Step 3.4.5.10.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.5.10.8

Add $1$ and $1$ .

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

Step 3.4.5.10.9

Multiply $- 1$ by $- 1$ .

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

Step 3.4.5.10.10

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

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

Step 3.4.5.11

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

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

Step 3.4.5.12

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

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

Step 3.4.5.13

Apply the distributive property.

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

Step 3.4.5.14

Multiply $a$ by $a$ .

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

Step 3.4.5.15

Rewrite using the commutative property of multiplication.

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

Step 3.4.6

The final answer is the combination of both solutions.

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

$x = 1$

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

$x = 1$

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

$x = 1$