Trigonometry Examples

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

Step 1

Solve the equation for $x$ .

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Step 1.1

Simplify $(x - a) (x - b)$ .

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Step 1.1.1

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

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Apply the distributive property.

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Multiply $x$ by $x$ .

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

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.3

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.4

Multiply $- 1$ by $- 1$ .

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

Step 1.1.2.5

Multiply $a$ by $1$ .

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

Step 1.2

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

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

Step 1.3

Use the quadratic formula to find the solutions.

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

Step 1.4

Substitute the values $a = 1$ , $b = - b - a$ , and $c = a b - c^{2}$ into the quadratic formula and solve for $x$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Apply the distributive property.

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

Step 1.5.1.2

Multiply $- - b$ .

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Step 1.5.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.1.2.2

Multiply $b$ by $1$ .

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

Step 1.5.1.3

Multiply $- - a$ .

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Step 1.5.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.1.3.2

Multiply $a$ by $1$ .

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

Step 1.5.1.4

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

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

Step 1.5.1.5

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

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Step 1.5.1.5.1

Apply the distributive property.

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

Step 1.5.1.5.2

Apply the distributive property.

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

Step 1.5.1.5.3

Apply the distributive property.

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

Step 1.5.1.6

Simplify and combine like terms.

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Step 1.5.1.6.1

Simplify each term.

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Step 1.5.1.6.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.6.1.2

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

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Step 1.5.1.6.1.2.1

Move $b$ .

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

Step 1.5.1.6.1.2.2

Multiply $b$ by $b$ .

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

Step 1.5.1.6.1.3

Multiply $- 1$ by $- 1$ .

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

Step 1.5.1.6.1.4

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

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

Step 1.5.1.6.1.5

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.5.1.6.1.7

Multiply $b$ by $1$ .

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

Step 1.5.1.6.1.8

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.6.1.9

Multiply $- 1$ by $- 1$ .

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

Step 1.5.1.6.1.10

Multiply $a$ by $1$ .

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

Step 1.5.1.6.1.11

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.6.1.12

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

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Step 1.5.1.6.1.12.1

Move $a$ .

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

Step 1.5.1.6.1.12.2

Multiply $a$ by $a$ .

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

Step 1.5.1.6.1.13

Multiply $- 1$ by $- 1$ .

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

Step 1.5.1.6.1.14

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

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

Step 1.5.1.6.2

Add $b a$ and $a b$ .

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Step 1.5.1.6.2.1

Reorder $b$ and $a$ .

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

Step 1.5.1.6.2.2

Add $a b$ and $a b$ .

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

Step 1.5.1.7

Multiply $- 4$ by $1$ .

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

Step 1.5.1.8

Apply the distributive property.

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

Step 1.5.1.9

Multiply $- 1$ by $- 4$ .

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

Step 1.5.1.10

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

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

Step 1.5.2

Multiply $2$ by $1$ .

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

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Apply the distributive property.

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

Step 1.6.1.2

Multiply $- - b$ .

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Step 1.6.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.2.2

Multiply $b$ by $1$ .

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

Step 1.6.1.3

Multiply $- - a$ .

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Step 1.6.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.3.2

Multiply $a$ by $1$ .

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

Step 1.6.1.4

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

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

Step 1.6.1.5

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

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Step 1.6.1.5.1

Apply the distributive property.

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

Step 1.6.1.5.2

Apply the distributive property.

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

Step 1.6.1.5.3

Apply the distributive property.

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

Step 1.6.1.6

Simplify and combine like terms.

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Step 1.6.1.6.1

Simplify each term.

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Step 1.6.1.6.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.6.1.2

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

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Step 1.6.1.6.1.2.1

Move $b$ .

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

Step 1.6.1.6.1.2.2

Multiply $b$ by $b$ .

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

Step 1.6.1.6.1.3

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.6.1.4

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

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

Step 1.6.1.6.1.5

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.6.1.7

Multiply $b$ by $1$ .

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

Step 1.6.1.6.1.8

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.6.1.9

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.6.1.10

Multiply $a$ by $1$ .

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

Step 1.6.1.6.1.11

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.6.1.12

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

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Step 1.6.1.6.1.12.1

Move $a$ .

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

Step 1.6.1.6.1.12.2

Multiply $a$ by $a$ .

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

Step 1.6.1.6.1.13

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.6.1.14

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

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

Step 1.6.1.6.2

Add $b a$ and $a b$ .

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Step 1.6.1.6.2.1

Reorder $b$ and $a$ .

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

Step 1.6.1.6.2.2

Add $a b$ and $a b$ .

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

Step 1.6.1.7

Multiply $- 4$ by $1$ .

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

Step 1.6.1.8

Apply the distributive property.

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

Step 1.6.1.9

Multiply $- 1$ by $- 4$ .

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

Step 1.6.1.10

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

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

Step 1.6.2

Multiply $2$ by $1$ .

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

Step 1.6.3

Change the $\pm$ to $+$ .

$x = \frac{b + a + \sqrt{b^{2} + a^{2} - 2 a b + 4 c^{2}}}{2}$

Step 1.7

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

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Step 1.7.1

Simplify the numerator.

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Step 1.7.1.1

Apply the distributive property.

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

Step 1.7.1.2

Multiply $- - b$ .

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Step 1.7.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.2.2

Multiply $b$ by $1$ .

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

Step 1.7.1.3

Multiply $- - a$ .

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Step 1.7.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.3.2

Multiply $a$ by $1$ .

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

Step 1.7.1.4

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

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

Step 1.7.1.5

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

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Step 1.7.1.5.1

Apply the distributive property.

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

Step 1.7.1.5.2

Apply the distributive property.

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

Step 1.7.1.5.3

Apply the distributive property.

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

Step 1.7.1.6

Simplify and combine like terms.

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Step 1.7.1.6.1

Simplify each term.

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Step 1.7.1.6.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.6.1.2

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

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Step 1.7.1.6.1.2.1

Move $b$ .

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

Step 1.7.1.6.1.2.2

Multiply $b$ by $b$ .

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

Step 1.7.1.6.1.3

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.6.1.4

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

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

Step 1.7.1.6.1.5

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.6.1.7

Multiply $b$ by $1$ .

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

Step 1.7.1.6.1.8

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.6.1.9

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.6.1.10

Multiply $a$ by $1$ .

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

Step 1.7.1.6.1.11

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.6.1.12

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

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Step 1.7.1.6.1.12.1

Move $a$ .

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

Step 1.7.1.6.1.12.2

Multiply $a$ by $a$ .

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

Step 1.7.1.6.1.13

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.6.1.14

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

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

Step 1.7.1.6.2

Add $b a$ and $a b$ .

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Step 1.7.1.6.2.1

Reorder $b$ and $a$ .

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

Step 1.7.1.6.2.2

Add $a b$ and $a b$ .

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

Step 1.7.1.7

Multiply $- 4$ by $1$ .

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

Step 1.7.1.8

Apply the distributive property.

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

Step 1.7.1.9

Multiply $- 1$ by $- 4$ .

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

Step 1.7.1.10

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

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

Step 1.7.2

Multiply $2$ by $1$ .

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

Step 1.7.3

Change the $\pm$ to $-$ .

$x = \frac{b + a - \sqrt{b^{2} + a^{2} - 2 a b + 4 c^{2}}}{2}$

Step 1.8

The final answer is the combination of both solutions.

$x = \frac{b + a + \sqrt{b^{2} + a^{2} - 2 a b + 4 c^{2}}}{2}$

$x = \frac{b + a - \sqrt{b^{2} + a^{2} - 2 a b + 4 c^{2}}}{2}$

$x = \frac{b + a + \sqrt{b^{2} + a^{2} - 2 a b + 4 c^{2}}}{2}$

$x = \frac{b + a - \sqrt{b^{2} + a^{2} - 2 a b + 4 c^{2}}}{2}$

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degrees of the variables in the equation violate the linear equation definition, which means that the equation is not a linear equation.

Not Linear