Pre-Algebra Examples

$9 x^{2} - 36 x = - 40$

Step 1

Add $40$ to both sides of the equation.

$9 x^{2} - 36 x + 40 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 9$ , $b = - 36$ , and $c = 40$ into the quadratic formula and solve for $x$ .

$\frac{36 \pm \sqrt{{(- 36)}^{2} - 4 \cdot (9 \cdot 40)}}{2 \cdot 9}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 36$ to the power of $2$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 9 \cdot 40}}{2 \cdot 9}$

Step 4.1.2

Multiply $- 4 \cdot 9 \cdot 40$ .

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Step 4.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{36 \pm \sqrt{1296 - 36 \cdot 40}}{2 \cdot 9}$

Step 4.1.2.2

Multiply $- 36$ by $40$ .

$x = \frac{36 \pm \sqrt{1296 - 1440}}{2 \cdot 9}$

Step 4.1.3

Subtract $1440$ from $1296$ .

$x = \frac{36 \pm \sqrt{- 144}}{2 \cdot 9}$

Step 4.1.4

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{36 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 9}$

Step 4.1.5

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{36 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 9}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{36 \pm i \cdot \sqrt{144}}{2 \cdot 9}$

Step 4.1.7

Rewrite $144$ as $12^{2}$ .

$x = \frac{36 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 9}$

Step 4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{36 \pm i \cdot 12}{2 \cdot 9}$

Step 4.1.9

Move $12$ to the left of $i$ .

$x = \frac{36 \pm 12 i}{2 \cdot 9}$

Step 4.2

Multiply $2$ by $9$ .

$x = \frac{36 \pm 12 i}{18}$

Step 4.3

Simplify $\frac{36 \pm 12 i}{18}$ .

$x = \frac{6 \pm 2 i}{3}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 36$ to the power of $2$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 9 \cdot 40}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot 40$ .

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Step 5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{36 \pm \sqrt{1296 - 36 \cdot 40}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $40$ .

$x = \frac{36 \pm \sqrt{1296 - 1440}}{2 \cdot 9}$

Step 5.1.3

Subtract $1440$ from $1296$ .

$x = \frac{36 \pm \sqrt{- 144}}{2 \cdot 9}$

Step 5.1.4

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{36 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 9}$

Step 5.1.5

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{36 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 9}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{36 \pm i \cdot \sqrt{144}}{2 \cdot 9}$

Step 5.1.7

Rewrite $144$ as $12^{2}$ .

$x = \frac{36 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 9}$

Step 5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{36 \pm i \cdot 12}{2 \cdot 9}$

Step 5.1.9

Move $12$ to the left of $i$ .

$x = \frac{36 \pm 12 i}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$x = \frac{36 \pm 12 i}{18}$

Step 5.3

Simplify $\frac{36 \pm 12 i}{18}$ .

$x = \frac{6 \pm 2 i}{3}$

Step 5.4

Change the $\pm$ to $+$ .

$x = \frac{6 + 2 i}{3}$

Step 5.5

Split the fraction $\frac{6 + 2 i}{3}$ into two fractions.

$x = \frac{6}{3} + \frac{2 i}{3}$

Step 5.6

Divide $6$ by $3$ .

$x = 2 + \frac{2 i}{3}$

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 $- 36$ to the power of $2$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 9 \cdot 40}}{2 \cdot 9}$

Step 6.1.2

Multiply $- 4 \cdot 9 \cdot 40$ .

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Step 6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{36 \pm \sqrt{1296 - 36 \cdot 40}}{2 \cdot 9}$

Step 6.1.2.2

Multiply $- 36$ by $40$ .

$x = \frac{36 \pm \sqrt{1296 - 1440}}{2 \cdot 9}$

Step 6.1.3

Subtract $1440$ from $1296$ .

$x = \frac{36 \pm \sqrt{- 144}}{2 \cdot 9}$

Step 6.1.4

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{36 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 9}$

Step 6.1.5

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{36 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 9}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{36 \pm i \cdot \sqrt{144}}{2 \cdot 9}$

Step 6.1.7

Rewrite $144$ as $12^{2}$ .

$x = \frac{36 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 9}$

Step 6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{36 \pm i \cdot 12}{2 \cdot 9}$

Step 6.1.9

Move $12$ to the left of $i$ .

$x = \frac{36 \pm 12 i}{2 \cdot 9}$

Step 6.2

Multiply $2$ by $9$ .

$x = \frac{36 \pm 12 i}{18}$

Step 6.3

Simplify $\frac{36 \pm 12 i}{18}$ .

$x = \frac{6 \pm 2 i}{3}$

Step 6.4

Change the $\pm$ to $-$ .

$x = \frac{6 - 2 i}{3}$

Step 6.5

Split the fraction $\frac{6 - 2 i}{3}$ into two fractions.

$x = \frac{6}{3} + \frac{- 2 i}{3}$

Step 6.6

Simplify each term.

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Step 6.6.1

Divide $6$ by $3$ .

$x = 2 + \frac{- 2 i}{3}$

Step 6.6.2

Move the negative in front of the fraction.

$x = 2 - \frac{2 i}{3}$

Step 7

The final answer is the combination of both solutions.

$x = 2 + \frac{2 i}{3}, 2 - \frac{2 i}{3}$