Pre-Algebra Examples

$9 y^{2} - 12 y + 13 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 9$ , $b = - 12$ , and $c = 13$ into the quadratic formula and solve for $y$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (9 \cdot 13)}}{2 \cdot 9}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 12$ to the power of $2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 9 \cdot 13}}{2 \cdot 9}$

Step 3.1.2

Multiply $- 4 \cdot 9 \cdot 13$ .

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Step 3.1.2.1

Multiply $- 4$ by $9$ .

$y = \frac{12 \pm \sqrt{144 - 36 \cdot 13}}{2 \cdot 9}$

Step 3.1.2.2

Multiply $- 36$ by $13$ .

$y = \frac{12 \pm \sqrt{144 - 468}}{2 \cdot 9}$

Step 3.1.3

Subtract $468$ from $144$ .

$y = \frac{12 \pm \sqrt{- 324}}{2 \cdot 9}$

Step 3.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$y = \frac{12 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 9}$

Step 3.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$y = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 9}$

Step 3.1.6

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

$y = \frac{12 \pm i \cdot \sqrt{324}}{2 \cdot 9}$

Step 3.1.7

Rewrite $324$ as $18^{2}$ .

$y = \frac{12 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 9}$

Step 3.1.8

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

$y = \frac{12 \pm i \cdot 18}{2 \cdot 9}$

Step 3.1.9

Move $18$ to the left of $i$ .

$y = \frac{12 \pm 18 i}{2 \cdot 9}$

Step 3.2

Multiply $2$ by $9$ .

$y = \frac{12 \pm 18 i}{18}$

Step 3.3

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

$y = \frac{2 \pm 3 i}{3}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 12$ to the power of $2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 9 \cdot 13}}{2 \cdot 9}$

Step 4.1.2

Multiply $- 4 \cdot 9 \cdot 13$ .

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Step 4.1.2.1

Multiply $- 4$ by $9$ .

$y = \frac{12 \pm \sqrt{144 - 36 \cdot 13}}{2 \cdot 9}$

Step 4.1.2.2

Multiply $- 36$ by $13$ .

$y = \frac{12 \pm \sqrt{144 - 468}}{2 \cdot 9}$

Step 4.1.3

Subtract $468$ from $144$ .

$y = \frac{12 \pm \sqrt{- 324}}{2 \cdot 9}$

Step 4.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$y = \frac{12 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 9}$

Step 4.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$y = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 9}$

Step 4.1.6

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

$y = \frac{12 \pm i \cdot \sqrt{324}}{2 \cdot 9}$

Step 4.1.7

Rewrite $324$ as $18^{2}$ .

$y = \frac{12 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 9}$

Step 4.1.8

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

$y = \frac{12 \pm i \cdot 18}{2 \cdot 9}$

Step 4.1.9

Move $18$ to the left of $i$ .

$y = \frac{12 \pm 18 i}{2 \cdot 9}$

Step 4.2

Multiply $2$ by $9$ .

$y = \frac{12 \pm 18 i}{18}$

Step 4.3

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

$y = \frac{2 \pm 3 i}{3}$

Step 4.4

Change the $\pm$ to $+$ .

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

Step 4.5

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

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

Step 4.6

Cancel the common factor of $3$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Divide $i$ by $1$ .

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

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 $- 12$ to the power of $2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 9 \cdot 13}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot 13$ .

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Step 5.1.2.1

Multiply $- 4$ by $9$ .

$y = \frac{12 \pm \sqrt{144 - 36 \cdot 13}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $13$ .

$y = \frac{12 \pm \sqrt{144 - 468}}{2 \cdot 9}$

Step 5.1.3

Subtract $468$ from $144$ .

$y = \frac{12 \pm \sqrt{- 324}}{2 \cdot 9}$

Step 5.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$y = \frac{12 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 9}$

Step 5.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$y = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 9}$

Step 5.1.6

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

$y = \frac{12 \pm i \cdot \sqrt{324}}{2 \cdot 9}$

Step 5.1.7

Rewrite $324$ as $18^{2}$ .

$y = \frac{12 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 9}$

Step 5.1.8

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

$y = \frac{12 \pm i \cdot 18}{2 \cdot 9}$

Step 5.1.9

Move $18$ to the left of $i$ .

$y = \frac{12 \pm 18 i}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$y = \frac{12 \pm 18 i}{18}$

Step 5.3

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

$y = \frac{2 \pm 3 i}{3}$

Step 5.4

Change the $\pm$ to $-$ .

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

Step 5.5

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

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

Step 5.6

Cancel the common factor of $- 3$ and $3$ .

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Step 5.6.1

Factor $3$ out of $- 3 i$ .

$y = \frac{2}{3} + \frac{3 (- i)}{3}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $3$ out of $3$ .

$y = \frac{2}{3} + \frac{3 (- i)}{3 (1)}$

Step 5.6.2.2

Cancel the common factor.

$y = \frac{2}{3} + \frac{3 (- i)}{3 \cdot 1}$

Step 5.6.2.3

Rewrite the expression.

$y = \frac{2}{3} + \frac{- i}{1}$

Step 5.6.2.4

Divide $- i$ by $1$ .

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

Step 6

The final answer is the combination of both solutions.

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