Pre-Algebra Examples

$2 x^{2} + 18 x + 5 = 3$

Step 1

Subtract $3$ from both sides of the equation.

$2 x^{2} + 18 x + 5 - 3 = 0$

Step 2

Subtract $3$ from $5$ .

$2 x^{2} + 18 x + 2 = 0$

Step 3

Factor $2$ out of $2 x^{2} + 18 x + 2$ .

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Step 3.1

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

$2 (x^{2}) + 18 x + 2 = 0$

Step 3.2

Factor $2$ out of $18 x$ .

$2 (x^{2}) + 2 (9 x) + 2 = 0$

Step 3.3

Factor $2$ out of $2$ .

$2 (x^{2}) + 2 (9 x) + 2 (1) = 0$

Step 3.4

Factor $2$ out of $2 (x^{2}) + 2 (9 x)$ .

$2 (x^{2} + 9 x) + 2 (1) = 0$

Step 3.5

Factor $2$ out of $2 (x^{2} + 9 x) + 2 (1)$ .

$2 (x^{2} + 9 x + 1) = 0$

Step 4

Divide each term in $2 (x^{2} + 9 x + 1) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (x^{2} + 9 x + 1) = 0$ by $2$ .

$\frac{2 (x^{2} + 9 x + 1)}{2} = \frac{0}{2}$

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.

$\frac{2 (x^{2} + 9 x + 1)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $x^{2} + 9 x + 1$ by $1$ .

$x^{2} + 9 x + 1 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$x^{2} + 9 x + 1 = 0$

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 $a = 1$ , $b = 9$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4}}{2 \cdot 1}$

Step 7.1.3

Subtract $4$ from $81$ .

$x = \frac{- 9 \pm \sqrt{77}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm \sqrt{77}}{2}$

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 $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 8.1.2

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

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4}}{2 \cdot 1}$

Step 8.1.3

Subtract $4$ from $81$ .

$x = \frac{- 9 \pm \sqrt{77}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm \sqrt{77}}{2}$

Step 8.3

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{77}}{2}$

Step 8.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{77}}{2}$

Step 8.5

Factor $- 1$ out of $\sqrt{77}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{77})}{2}$

Step 8.6

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{77})$ .

$x = \frac{- 1 (9 - \sqrt{77})}{2}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{77}}{2}$

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 $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 9.1.2

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

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4}}{2 \cdot 1}$

Step 9.1.3

Subtract $4$ from $81$ .

$x = \frac{- 9 \pm \sqrt{77}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm \sqrt{77}}{2}$

Step 9.3

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{77}}{2}$

Step 9.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{77}}{2}$

Step 9.5

Factor $- 1$ out of $- \sqrt{77}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{77})}{2}$

Step 9.6

Factor $- 1$ out of $- 1 (9) - (\sqrt{77})$ .

$x = \frac{- 1 (9 + \sqrt{77})}{2}$

Step 9.7

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{77}}{2}$

Step 10

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{77}}{2}, - \frac{9 + \sqrt{77}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{77}}{2}, - \frac{9 + \sqrt{77}}{2}$

Decimal Form:

$x = - 0.11251780 \dots, - 8.88748219 \dots$