Pre-Algebra Examples

$x (3 x - 2) = x^{2} + 5$

Step 1

Simplify $x (3 x - 2)$ .

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Step 1.1

Rewrite.

$0 + 0 + x (3 x - 2) = x^{2} + 5$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$x (3 x) + x \cdot - 2 = x^{2} + 5$

Step 1.2.2

Reorder.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot - 2 = x^{2} + 5$

Step 1.2.2.2

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

$3 x \cdot x - 2 \cdot x = x^{2} + 5$

Step 1.3

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

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Step 1.3.1

Move $x$ .

$3 (x \cdot x) - 2 \cdot x = x^{2} + 5$

Step 1.3.2

Multiply $x$ by $x$ .

$3 x^{2} - 2 \cdot x = x^{2} + 5$

$3 x^{2} - 2 x = x^{2} + 5$

Step 2

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

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$3 x^{2} - 2 x - x^{2} = 5$

Step 2.2

Subtract $x^{2}$ from $3 x^{2}$ .

$2 x^{2} - 2 x = 5$

Step 3

Subtract $5$ from both sides of the equation.

$2 x^{2} - 2 x - 5 = 0$

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 $a = 2$ , $b = - 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot - 5)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 5}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot 2}$

Step 6.1.3

Add $4$ and $40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot 2}$

Step 6.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 6.1.4.1

Factor $4$ out of $44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Step 6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{4}$

Step 6.3

Simplify $\frac{2 \pm 2 \sqrt{11}}{4}$ .

$x = \frac{1 \pm \sqrt{11}}{2}$

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 $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 5}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot 2}$

Step 7.1.3

Add $4$ and $40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot 2}$

Step 7.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 7.1.4.1

Factor $4$ out of $44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{4}$

Step 7.3

Simplify $\frac{2 \pm 2 \sqrt{11}}{4}$ .

$x = \frac{1 \pm \sqrt{11}}{2}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{11}}{2}$

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 8.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 8.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 5}}{2 \cdot 2}$

Step 8.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot 2}$

Step 8.1.3

Add $4$ and $40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot 2}$

Step 8.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 8.1.4.1

Factor $4$ out of $44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{4}$

Step 8.3

Simplify $\frac{2 \pm 2 \sqrt{11}}{4}$ .

$x = \frac{1 \pm \sqrt{11}}{2}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{11}}{2}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{11}}{2}, \frac{1 - \sqrt{11}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{11}}{2}, \frac{1 - \sqrt{11}}{2}$

Decimal Form:

$x = 2.15831239 \dots, - 1.15831239 \dots$