Pre-Algebra Examples

$x (4 x - 2) = 66$

Step 1

Simplify $x (4 x - 2)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x (4 x) + x \cdot - 2 = 66$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$4 x \cdot x + x \cdot - 2 = 66$

Step 1.1.2.2

Move $- 2$ to the left of $x$ .

$4 x \cdot x - 2 \cdot x = 66$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$4 (x \cdot x) - 2 \cdot x = 66$

Step 1.2.2

Multiply $x$ by $x$ .

$4 x^{2} - 2 \cdot x = 66$

$4 x^{2} - 2 x = 66$

Step 2

Subtract $66$ from both sides of the equation.

$4 x^{2} - 2 x - 66 = 0$

Step 3

Factor $2$ out of $4 x^{2} - 2 x - 66$ .

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Step 3.1

Factor $2$ out of $4 x^{2}$ .

$2 (2 x^{2}) - 2 x - 66 = 0$

Step 3.2

Factor $2$ out of $- 2 x$ .

$2 (2 x^{2}) + 2 (- x) - 66 = 0$

Step 3.3

Factor $2$ out of $- 66$ .

$2 (2 x^{2}) + 2 (- x) + 2 (- 33) = 0$

Step 3.4

Factor $2$ out of $2 (2 x^{2}) + 2 (- x)$ .

$2 (2 x^{2} - x) + 2 (- 33) = 0$

Step 3.5

Factor $2$ out of $2 (2 x^{2} - x) + 2 (- 33)$ .

$2 (2 x^{2} - x - 33) = 0$

Step 4

Divide each term in $2 (2 x^{2} - x - 33) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (2 x^{2} - x - 33) = 0$ by $2$ .

$\frac{2 (2 x^{2} - x - 33)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (2 x^{2} - x - 33)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $2 x^{2} - x - 33$ by $1$ .

$2 x^{2} - x - 33 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$2 x^{2} - x - 33 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 2$ , $b = - 1$ , and $c = - 33$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot - 33)}}{2 \cdot 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 33}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 33$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 33}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 33$ .

$x = \frac{1 \pm \sqrt{1 + 264}}{2 \cdot 2}$

Step 7.1.3

Add $1$ and $264$ .

$x = \frac{1 \pm \sqrt{265}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{265}}{4}$

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 33}}{2 \cdot 2}$

Step 8.1.2

Multiply $- 4 \cdot 2 \cdot - 33$ .

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Step 8.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 33}}{2 \cdot 2}$

Step 8.1.2.2

Multiply $- 8$ by $- 33$ .

$x = \frac{1 \pm \sqrt{1 + 264}}{2 \cdot 2}$

Step 8.1.3

Add $1$ and $264$ .

$x = \frac{1 \pm \sqrt{265}}{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{265}}{4}$

Step 8.3

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{265}}{4}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 33}}{2 \cdot 2}$

Step 9.1.2

Multiply $- 4 \cdot 2 \cdot - 33$ .

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Step 9.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 33}}{2 \cdot 2}$

Step 9.1.2.2

Multiply $- 8$ by $- 33$ .

$x = \frac{1 \pm \sqrt{1 + 264}}{2 \cdot 2}$

Step 9.1.3

Add $1$ and $264$ .

$x = \frac{1 \pm \sqrt{265}}{2 \cdot 2}$

Step 9.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{265}}{4}$

Step 9.3

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{265}}{4}$

Step 10

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{265}}{4}, \frac{1 - \sqrt{265}}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{265}}{4}, \frac{1 - \sqrt{265}}{4}$

Decimal Form:

$x = 4.31970514 \dots, - 3.81970514 \dots$