Pre-Algebra Examples

$- 2 x (x - 4) = - 16 + x$

Step 1

Simplify $- 2 x (x - 4)$ .

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Step 1.1

Rewrite.

$0 + 0 - 2 x (x - 4) = - 16 + x$

Step 1.2

Simplify by adding zeros.

$- 2 x (x - 4) = - 16 + x$

Step 1.3

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 4 = - 16 + x$

Step 1.4

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

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Step 1.4.1

Move $x$ .

$- 2 (x \cdot x) - 2 x \cdot - 4 = - 16 + x$

Step 1.4.2

Multiply $x$ by $x$ .

$- 2 x^{2} - 2 x \cdot - 4 = - 16 + x$

Step 1.5

Multiply $- 4$ by $- 2$ .

$- 2 x^{2} + 8 x = - 16 + x$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $x$ from both sides of the equation.

$- 2 x^{2} + 8 x - x = - 16$

Step 2.2

Subtract $x$ from $8 x$ .

$- 2 x^{2} + 7 x = - 16$

Step 3

Add $16$ to both sides of the equation.

$- 2 x^{2} + 7 x + 16 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 2$ , $b = 7$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (- 2 \cdot 16)}}{2 \cdot - 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot 16}}{2 \cdot - 2}$

Step 6.1.2

Multiply $- 4 \cdot - 2 \cdot 16$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot 16}}{2 \cdot - 2}$

Step 6.1.2.2

Multiply $8$ by $16$ .

$x = \frac{- 7 \pm \sqrt{49 + 128}}{2 \cdot - 2}$

Step 6.1.3

Add $49$ and $128$ .

$x = \frac{- 7 \pm \sqrt{177}}{2 \cdot - 2}$

Step 6.2

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{177}}{- 4}$

Step 6.3

Simplify $\frac{- 7 \pm \sqrt{177}}{- 4}$ .

$x = \frac{7 \pm \sqrt{177}}{4}$

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 $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot 16}}{2 \cdot - 2}$

Step 7.1.2

Multiply $- 4 \cdot - 2 \cdot 16$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot 16}}{2 \cdot - 2}$

Step 7.1.2.2

Multiply $8$ by $16$ .

$x = \frac{- 7 \pm \sqrt{49 + 128}}{2 \cdot - 2}$

Step 7.1.3

Add $49$ and $128$ .

$x = \frac{- 7 \pm \sqrt{177}}{2 \cdot - 2}$

Step 7.2

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{177}}{- 4}$

Step 7.3

Simplify $\frac{- 7 \pm \sqrt{177}}{- 4}$ .

$x = \frac{7 \pm \sqrt{177}}{4}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{177}}{4}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot 16}}{2 \cdot - 2}$

Step 8.1.2

Multiply $- 4 \cdot - 2 \cdot 16$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot 16}}{2 \cdot - 2}$

Step 8.1.2.2

Multiply $8$ by $16$ .

$x = \frac{- 7 \pm \sqrt{49 + 128}}{2 \cdot - 2}$

Step 8.1.3

Add $49$ and $128$ .

$x = \frac{- 7 \pm \sqrt{177}}{2 \cdot - 2}$

Step 8.2

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{177}}{- 4}$

Step 8.3

Simplify $\frac{- 7 \pm \sqrt{177}}{- 4}$ .

$x = \frac{7 \pm \sqrt{177}}{4}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{177}}{4}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{177}}{4}, \frac{7 - \sqrt{177}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + \sqrt{177}}{4}, \frac{7 - \sqrt{177}}{4}$

Decimal Form:

$x = 5.07603367 \dots, - 1.57603367 \dots$