Examples

$x^{2} + 7 x - 8 = 1$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $1$ from both sides of the equation.

$x^{2} + 7 x - 8 - 1 = 0$

Step 1.2

Subtract $1$ from $- 8$ .

$x^{2} + 7 x - 9 = 0$

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 $a = 1$ , $b = 7$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 9}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 7 \pm \sqrt{49 + 36}}{2 \cdot 1}$

Step 4.1.3

Add $49$ and $36$ .

$x = \frac{- 7 \pm \sqrt{85}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{85}}{2}$

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

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 9}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 7 \pm \sqrt{49 + 36}}{2 \cdot 1}$

Step 5.1.3

Add $49$ and $36$ .

$x = \frac{- 7 \pm \sqrt{85}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{85}}{2}$

Step 5.3

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{85}}{2}$

Step 5.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{85}}{2}$

Step 5.5

Factor $- 1$ out of $\sqrt{85}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{85})}{2}$

Step 5.6

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{85})$ .

$x = \frac{- 1 (7 - \sqrt{85})}{2}$

Step 5.7

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{85}}{2}$

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

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 9}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 7 \pm \sqrt{49 + 36}}{2 \cdot 1}$

Step 6.1.3

Add $49$ and $36$ .

$x = \frac{- 7 \pm \sqrt{85}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{85}}{2}$

Step 6.3

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{85}}{2}$

Step 6.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{85}}{2}$

Step 6.5

Factor $- 1$ out of $- \sqrt{85}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{85})}{2}$

Step 6.6

Factor $- 1$ out of $- 1 (7) - (\sqrt{85})$ .

$x = \frac{- 1 (7 + \sqrt{85})}{2}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{85}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{85}}{2}, - \frac{7 + \sqrt{85}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{85}}{2}, - \frac{7 + \sqrt{85}}{2}$

Decimal Form:

$x = 1.10977222 \dots, - 8.10977222 \dots$

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