Pre-Algebra Examples

$x (2 x - 1) = 17$

Step 1

Simplify $x (2 x - 1)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x (2 x) + x \cdot - 1 = 17$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 1 = 17$

Step 1.1.2.2

Move $- 1$ to the left of $x$ .

$2 x \cdot x - 1 \cdot x = 17$

Step 1.2

Simplify each term.

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Step 1.2.1

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

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Step 1.2.1.1

Move $x$ .

$2 (x \cdot x) - 1 \cdot x = 17$

Step 1.2.1.2

Multiply $x$ by $x$ .

$2 x^{2} - 1 \cdot x = 17$

Step 1.2.2

Rewrite $- 1 x$ as $- x$ .

$2 x^{2} - x = 17$

Step 2

Subtract $17$ from both sides of the equation.

$2 x^{2} - x - 17 = 0$

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 $a = 2$ , $b = - 1$ , and $c = - 17$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot - 17)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 17}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 17$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 17}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 17$ .

$x = \frac{1 \pm \sqrt{1 + 136}}{2 \cdot 2}$

Step 5.1.3

Add $1$ and $136$ .

$x = \frac{1 \pm \sqrt{137}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{137}}{4}$

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 17}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 17$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 17}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 17$ .

$x = \frac{1 \pm \sqrt{1 + 136}}{2 \cdot 2}$

Step 6.1.3

Add $1$ and $136$ .

$x = \frac{1 \pm \sqrt{137}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{137}}{4}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{137}}{4}$

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 17}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 17$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 17}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 17$ .

$x = \frac{1 \pm \sqrt{1 + 136}}{2 \cdot 2}$

Step 7.1.3

Add $1$ and $136$ .

$x = \frac{1 \pm \sqrt{137}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{137}}{4}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{137}}{4}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{137}}{4}, \frac{1 - \sqrt{137}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{137}}{4}, \frac{1 - \sqrt{137}}{4}$

Decimal Form:

$x = 3.17617497 \dots, - 2.67617497 \dots$