Pre-Algebra Examples

$2 x - 2 + x^{2} - 2 x + x + 9 + x + 4 = 42$

Step 1

Simplify $2 x - 2 + x^{2} - 2 x + x + 9 + x + 4$ .

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Step 1.1

Combine the opposite terms in $2 x - 2 + x^{2} - 2 x + x + 9 + x + 4$ .

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Step 1.1.1

Subtract $2 x$ from $2 x$ .

$- 2 + x^{2} + 0 + x + 9 + x + 4 = 42$

Step 1.1.2

Add $- 2 + x^{2}$ and $0$ .

$- 2 + x^{2} + x + 9 + x + 4 = 42$

Step 1.2

Add $- 2$ and $9$ .

$x^{2} + x + 7 + x + 4 = 42$

Step 1.3

Add $x$ and $x$ .

$x^{2} + 2 x + 7 + 4 = 42$

Step 1.4

Add $7$ and $4$ .

$x^{2} + 2 x + 11 = 42$

Step 2

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

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Step 2.1

Subtract $42$ from both sides of the equation.

$x^{2} + 2 x + 11 - 42 = 0$

Step 2.2

Subtract $42$ from $11$ .

$x^{2} + 2 x - 31 = 0$

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 $a = 1$ , $b = 2$ , and $c = - 31$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 31)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 31}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 31$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 31}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 31$ .

$x = \frac{- 2 \pm \sqrt{4 + 124}}{2 \cdot 1}$

Step 5.1.3

Add $4$ and $124$ .

$x = \frac{- 2 \pm \sqrt{128}}{2 \cdot 1}$

Step 5.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 5.1.4.1

Factor $64$ out of $128$ .

$x = \frac{- 2 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 2 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 8 \sqrt{2}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 8 \sqrt{2}}{2}$

Step 5.3

Simplify $\frac{- 2 \pm 8 \sqrt{2}}{2}$ .

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

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 $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 31}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 31$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 31}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 31$ .

$x = \frac{- 2 \pm \sqrt{4 + 124}}{2 \cdot 1}$

Step 6.1.3

Add $4$ and $124$ .

$x = \frac{- 2 \pm \sqrt{128}}{2 \cdot 1}$

Step 6.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 6.1.4.1

Factor $64$ out of $128$ .

$x = \frac{- 2 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Step 6.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 2 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 8 \sqrt{2}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 8 \sqrt{2}}{2}$

Step 6.3

Simplify $\frac{- 2 \pm 8 \sqrt{2}}{2}$ .

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

Step 6.4

Change the $\pm$ to $+$ .

$x = - 1 + 4 \sqrt{2}$

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

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 31}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 31$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 31}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 31$ .

$x = \frac{- 2 \pm \sqrt{4 + 124}}{2 \cdot 1}$

Step 7.1.3

Add $4$ and $124$ .

$x = \frac{- 2 \pm \sqrt{128}}{2 \cdot 1}$

Step 7.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 7.1.4.1

Factor $64$ out of $128$ .

$x = \frac{- 2 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 2 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 8 \sqrt{2}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 8 \sqrt{2}}{2}$

Step 7.3

Simplify $\frac{- 2 \pm 8 \sqrt{2}}{2}$ .

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

Step 7.4

Change the $\pm$ to $-$ .

$x = - 1 - 4 \sqrt{2}$

Step 8

The final answer is the combination of both solutions.

$x = - 1 + 4 \sqrt{2}, - 1 - 4 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 1 + 4 \sqrt{2}, - 1 - 4 \sqrt{2}$

Decimal Form:

$x = 4.65685424 \dots, - 6.65685424 \dots$