Finite Math Examples

$b^{2} x - c^{2} x + c^{2} b - b x^{2} + c x^{2} - b^{2} c$

Step 1

Regroup terms.

$- c^{2} x + c x^{2} + b^{2} x + c^{2} b - b x^{2} - b^{2} c$

Step 2

Factor $- c x$ out of $- c^{2} x + c x^{2}$ .

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Step 2.1

Factor $- c x$ out of $- c^{2} x$ .

$- c x (c) + c x^{2} + b^{2} x + c^{2} b - b x^{2} - b^{2} c$

Step 2.2

Factor $- c x$ out of $c x^{2}$ .

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

Step 2.3

Factor $- c x$ out of $- c x (c) - c x (- 1 x)$ .

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

Step 3

Rewrite $- 1 x$ as $- x$ .

$- c x (c - x) + b^{2} x + c^{2} b - b x^{2} - b^{2} c$

Step 4

Factor $b$ out of $b^{2} x + c^{2} b - b x^{2} - b^{2} c$ .

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Step 4.1

Factor $b$ out of $b^{2} x$ .

$- c x (c - x) + b (b x) + c^{2} b - b x^{2} - b^{2} c$

Step 4.2

Factor $b$ out of $c^{2} b$ .

$- c x (c - x) + b (b x) + b c^{2} - b x^{2} - b^{2} c$

Step 4.3

Factor $b$ out of $- b x^{2}$ .

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

Step 4.4

Factor $b$ out of $- b^{2} c$ .

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

Step 4.5

Factor $b$ out of $b (b x) + b c^{2}$ .

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

Step 4.6

Factor $b$ out of $b (b x + c^{2}) + b (- 1 x^{2})$ .

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

Step 4.7

Factor $b$ out of $b (b x + c^{2} - 1 x^{2}) + b (- b c)$ .

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

Step 5

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

$- c x (c - x) + b (b x + c^{2} - x^{2} - b c)$

Step 6

Factor $- 1$ out of $- c x (c - x) + b (b x + c^{2} - x^{2} - b c)$ .

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Step 6.1

Reorder the expression.

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Step 6.1.1

Move $- x^{2}$ .

$- c x (c - x) + b (b x + c^{2} - b c - x^{2})$

Step 6.1.2

Move $c^{2}$ .

$- c x (c - x) + b (b x - b c + c^{2} - x^{2})$

Step 6.1.3

Reorder $b x$ and $- b c$ .

$- c x (c - x) + b (- b c + b x + c^{2} - x^{2})$

Step 6.1.4

Reorder $- c x (c - x)$ and $b (- b c + b x + c^{2} - x^{2})$ .

$b (- b c + b x + c^{2} - x^{2}) - c x (c - x)$

Step 6.2

Factor $- 1$ out of $b (- b c + b x + c^{2} - x^{2})$ .

$- (- b (- b c + b x + c^{2} - x^{2})) - c x (c - x)$

Step 6.3

Factor $- 1$ out of $- c x (c - x)$ .

$- (- b (- b c + b x + c^{2} - x^{2})) - (c x (c - x))$

Step 6.4

Factor $- 1$ out of $- (- b (- b c + b x + c^{2} - x^{2})) - (c x (c - x))$ .

$- (- b (- b c + b x + c^{2} - x^{2}) + c x (c - x))$

Step 7

Apply the distributive property.

$- (- b (- b c) - b (b x) - b c^{2} - b (- x^{2}) + c x (c - x))$

Step 8

Simplify.

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Step 8.1

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

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Step 8.1.1

Move $b$ .

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

Step 8.1.2

Multiply $b$ by $b$ .

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

Step 8.2

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

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Step 8.2.1

Move $b$ .

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

Step 8.2.2

Multiply $b$ by $b$ .

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

Step 8.3

Rewrite using the commutative property of multiplication.

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

Step 9

Simplify each term.

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Step 9.1

Rewrite using the commutative property of multiplication.

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

Step 9.2

Multiply $- 1$ by $- 1$ .

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

Step 9.3

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

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

Step 9.4

Multiply $- 1$ by $- 1$ .

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

Step 9.5

Multiply $b$ by $1$ .

$- (b^{2} c - b^{2} x - b c^{2} + b x^{2} + c x (c - x))$

Step 10

Apply the distributive property.

$- (b^{2} c - b^{2} x - b c^{2} + b x^{2} + c x c + c x (- x))$

Step 11

Multiply $c$ by $c$ by adding the exponents.

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Step 11.1

Move $c$ .

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

Step 11.2

Multiply $c$ by $c$ .

$- (b^{2} c - b^{2} x - b c^{2} + b x^{2} + c^{2} x + c x (- x))$

Step 12

Rewrite using the commutative property of multiplication.

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

Step 13

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

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Step 13.1

Move $x$ .

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

Step 13.2

Multiply $x$ by $x$ .

$- (b^{2} c - b^{2} x - b c^{2} + b x^{2} + c^{2} x - c x^{2})$

Step 14

Use the quadratic formula to find the roots for $b^{2} c - b^{2} x - b c^{2} + b x^{2} + c^{2} x - c x^{2} = 0$

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Step 14.1

Use the quadratic formula to find the solutions.

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

Step 14.2

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

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

Step 14.3

Simplify the numerator.

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Step 14.3.1

Apply the distributive property.

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

Step 14.3.2

Multiply $- - c^{2}$ .

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Step 14.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 14.3.2.2

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

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

Step 14.3.3

Rewrite ${(- c^{2} + x^{2})}^{2}$ as $(- c^{2} + x^{2}) (- c^{2} + x^{2})$ .

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

Step 14.3.4

Expand $(- c^{2} + x^{2}) (- c^{2} + x^{2})$ using the FOIL Method.

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Step 14.3.4.1

Apply the distributive property.

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

Step 14.3.4.2

Apply the distributive property.

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

Step 14.3.4.3

Apply the distributive property.

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

Step 14.3.5

Simplify and combine like terms.

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Step 14.3.5.1

Simplify each term.

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Step 14.3.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 14.3.5.1.2

Multiply $c^{2}$ by $c^{2}$ by adding the exponents.

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Step 14.3.5.1.2.1

Move $c^{2}$ .

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

Step 14.3.5.1.2.2

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

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

Step 14.3.5.1.2.3

Add $2$ and $2$ .

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

Step 14.3.5.1.3

Multiply $- 1$ by $- 1$ .

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

Step 14.3.5.1.4

Multiply $c^{4}$ by $1$ .

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

Step 14.3.5.1.5

Rewrite using the commutative property of multiplication.

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

Step 14.3.5.1.6

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 14.3.5.1.6.1

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

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

Step 14.3.5.1.6.2

Add $2$ and $2$ .

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

Step 14.3.5.2

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

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Step 14.3.5.2.1

Move $x^{2}$ .

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

Step 14.3.5.2.2

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

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

Step 14.3.6

Apply the distributive property.

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

Step 14.3.7

Multiply $- 1$ by $- 4$ .

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

Step 14.3.8

Expand $(- 4 c + 4 x) (c^{2} x - c x^{2})$ using the FOIL Method.

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Step 14.3.8.1

Apply the distributive property.

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

Step 14.3.8.2

Apply the distributive property.

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

Step 14.3.8.3

Apply the distributive property.

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

Step 14.3.9

Simplify and combine like terms.

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Step 14.3.9.1

Simplify each term.

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Step 14.3.9.1.1

Multiply $c$ by $c^{2}$ by adding the exponents.

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Step 14.3.9.1.1.1

Move $c^{2}$ .

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

Step 14.3.9.1.1.2

Multiply $c^{2}$ by $c$ .

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Step 14.3.9.1.1.2.1

Raise $c$ to the power of $1$ .

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

Step 14.3.9.1.1.2.2

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

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

Step 14.3.9.1.1.3

Add $2$ and $1$ .

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

Step 14.3.9.1.2

Multiply $c$ by $c$ by adding the exponents.

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Step 14.3.9.1.2.1

Move $c$ .

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 2 c^{2} x^{2} + x^{4} - 4 c^{3} x - 4 (c \cdot c) (- 1 x^{2}) + 4 x (c^{2} x) + 4 x (- c x^{2})}}{2 (c - x)}$

Step 14.3.9.1.2.2

Multiply $c$ by $c$ .

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 2 c^{2} x^{2} + x^{4} - 4 c^{3} x - 4 c^{2} (- 1 x^{2}) + 4 x (c^{2} x) + 4 x (- c x^{2})}}{2 (c - x)}$

Step 14.3.9.1.3

Rewrite using the commutative property of multiplication.

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 2 c^{2} x^{2} + x^{4} - 4 c^{3} x - 4 \cdot (- 1 c^{2} x^{2}) + 4 x (c^{2} x) + 4 x (- c x^{2})}}{2 (c - x)}$

Step 14.3.9.1.4

Multiply $- 4$ by $- 1$ .

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

Step 14.3.9.1.5

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

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Step 14.3.9.1.5.1

Move $x$ .

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

Step 14.3.9.1.5.2

Multiply $x$ by $x$ .

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

Step 14.3.9.1.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 14.3.9.1.6.1

Move $x^{2}$ .

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

Step 14.3.9.1.6.2

Multiply $x^{2}$ by $x$ .

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Step 14.3.9.1.6.2.1

Raise $x$ to the power of $1$ .

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

Step 14.3.9.1.6.2.2

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

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 2 c^{2} x^{2} + x^{4} - 4 c^{3} x + 4 c^{2} x^{2} + 4 x^{2} c^{2} + 4 x^{2 + 1} (- c)}}{2 (c - x)}$

Step 14.3.9.1.6.3

Add $2$ and $1$ .

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

Step 14.3.9.1.7

Rewrite using the commutative property of multiplication.

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 2 c^{2} x^{2} + x^{4} - 4 c^{3} x + 4 c^{2} x^{2} + 4 x^{2} c^{2} + 4 \cdot (- 1 x^{3} c)}}{2 (c - x)}$

Step 14.3.9.1.8

Multiply $4$ by $- 1$ .

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

Step 14.3.9.2

Add $4 c^{2} x^{2}$ and $4 x^{2} c^{2}$ .

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Step 14.3.9.2.1

Move $x^{2}$ .

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

Step 14.3.9.2.2

Add $4 c^{2} x^{2}$ and $4 c^{2} x^{2}$ .

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 2 c^{2} x^{2} + x^{4} - 4 c^{3} x + 8 c^{2} x^{2} - 4 x^{3} c}}{2 (c - x)}$

Step 14.3.10

Add $- 2 c^{2} x^{2}$ and $8 c^{2} x^{2}$ .

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} + x^{4} - 4 c^{3} x + 6 c^{2} x^{2} - 4 x^{3} c}}{2 (c - x)}$

Step 14.3.11

Rewrite $c^{4} + x^{4} - 4 c^{3} x + 6 c^{2} x^{2} - 4 x^{3} c$ in a factored form.

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Step 14.3.11.1

Move $x^{3}$ .

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} + x^{4} - 4 c^{3} x + 6 c^{2} x^{2} - 4 c x^{3}}}{2 (c - x)}$

Step 14.3.11.2

Move $x^{4}$ .

$b = \frac{c^{2} - x^{2} \pm \sqrt{c^{4} - 4 c^{3} x + 6 c^{2} x^{2} - 4 c x^{3} + x^{4}}}{2 (c - x)}$

Step 14.3.11.3

Factor $c^{4} - 4 c^{3} x + 6 c^{2} x^{2} - 4 c x^{3} + x^{4}$ using the binomial theorem.

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

Step 14.3.12

Rewrite ${(c - x)}^{4}$ as ${({(c - x)}^{2})}^{2}$ .

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

Step 14.3.13

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

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

Step 14.3.14

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

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

Step 14.3.15

Expand $(c - x) (c - x)$ using the FOIL Method.

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Step 14.3.15.1

Apply the distributive property.

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

Step 14.3.15.2

Apply the distributive property.

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

Step 14.3.15.3

Apply the distributive property.

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

Step 14.3.16

Simplify and combine like terms.

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Step 14.3.16.1

Simplify each term.

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Step 14.3.16.1.1

Multiply $c$ by $c$ .

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

Step 14.3.16.1.2

Rewrite using the commutative property of multiplication.

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

Step 14.3.16.1.3

Rewrite using the commutative property of multiplication.

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

Step 14.3.16.1.4

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

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Step 14.3.16.1.4.1

Move $x$ .

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

Step 14.3.16.1.4.2

Multiply $x$ by $x$ .

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

Step 14.3.16.1.5

Multiply $- 1$ by $- 1$ .

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

Step 14.3.16.1.6

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

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

Step 14.3.16.2

Subtract $x c$ from $- c x$ .

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Step 14.3.16.2.1

Move $x$ .

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

Step 14.3.16.2.2

Subtract $c x$ from $- c x$ .

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

Step 15

Find the factors from the roots, then multiply the factors together.

$(- c + x) (b - \frac{c^{2} - x^{2} + c^{2} - 2 c x + x^{2}}{2 (c - x)}) (b - (\frac{c^{2} - x^{2} - (c^{2} - 2 c x + x^{2})}{2 (c - x)}))$

Step 16

Simplify the factored form.

$(- c + x) (b - \frac{c^{2} - x^{2} + c^{2} - 2 c x + x^{2}}{2 (c - x)}) (b + \frac{- c^{2} + x^{2} + c^{2} - 2 c x + x^{2}}{2 (c - x)})$