Pre-Algebra Examples

${(- x)}^{2} + (- 11 + x) = 40$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $- x$ .

${(- 1)}^{2} x^{2} - 11 + x = 40$

Step 1.2

Raise $- 1$ to the power of $2$ .

$1 x^{2} - 11 + x = 40$

Step 1.3

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

$x^{2} - 11 + x = 40$

Step 2

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

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Step 2.1

Subtract $40$ from both sides of the equation.

$x^{2} - 11 + x - 40 = 0$

Step 2.2

Subtract $40$ from $- 11$ .

$x^{2} + x - 51 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 1$ , and $c = - 51$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 51)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 51}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 51$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 51}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 51$ .

$x = \frac{- 1 \pm \sqrt{1 + 204}}{2 \cdot 1}$

Step 5.1.3

Add $1$ and $204$ .

$x = \frac{- 1 \pm \sqrt{205}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{205}}{2}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 51}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 51$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 51}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 51$ .

$x = \frac{- 1 \pm \sqrt{1 + 204}}{2 \cdot 1}$

Step 6.1.3

Add $1$ and $204$ .

$x = \frac{- 1 \pm \sqrt{205}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{205}}{2}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{205}}{2}$

Step 6.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{205}}{2}$

Step 6.5

Factor $- 1$ out of $\sqrt{205}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{205})}{2}$

Step 6.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{205})$ .

$x = \frac{- 1 (1 - \sqrt{205})}{2}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{205}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 51}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 51$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 51}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 51$ .

$x = \frac{- 1 \pm \sqrt{1 + 204}}{2 \cdot 1}$

Step 7.1.3

Add $1$ and $204$ .

$x = \frac{- 1 \pm \sqrt{205}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{205}}{2}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{205}}{2}$

Step 7.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{205}}{2}$

Step 7.5

Factor $- 1$ out of $- \sqrt{205}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{205})}{2}$

Step 7.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{205})$ .

$x = \frac{- 1 (1 + \sqrt{205})}{2}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{205}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{205}}{2}, - \frac{1 + \sqrt{205}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{205}}{2}, - \frac{1 + \sqrt{205}}{2}$

Decimal Form:

$x = 6.65891053 \dots, - 7.65891053 \dots$