Pre-Algebra Examples

$\frac{1}{3} x^{2} + \frac{1}{15} x - 1 = 0$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{3}$ and $x^{2}$ .

$\frac{x^{2}}{3} + \frac{1}{15} x - 1 = 0$

Step 1.2

Combine $\frac{1}{15}$ and $x$ .

$\frac{x^{2}}{3} + \frac{x}{15} - 1 = 0$

Step 2

Multiply through by the least common denominator $15$ , then simplify.

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Step 2.1

Apply the distributive property.

$15 (\frac{x^{2}}{3}) + 15 (\frac{x}{15}) + 15 \cdot - 1 = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Factor $3$ out of $15$ .

$3 (5) (\frac{x^{2}}{3}) + 15 (\frac{x}{15}) + 15 \cdot - 1 = 0$

Step 2.2.1.2

Cancel the common factor.

$3 \cdot (5 (\frac{x^{2}}{3})) + 15 (\frac{x}{15}) + 15 \cdot - 1 = 0$

Step 2.2.1.3

Rewrite the expression.

$5 x^{2} + 15 (\frac{x}{15}) + 15 \cdot - 1 = 0$

Step 2.2.2

Cancel the common factor of $15$ .

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Step 2.2.2.1

Cancel the common factor.

$5 x^{2} + 15 (\frac{x}{15}) + 15 \cdot - 1 = 0$

Step 2.2.2.2

Rewrite the expression.

$5 x^{2} + x + 15 \cdot - 1 = 0$

Step 2.2.3

Multiply $15$ by $- 1$ .

$5 x^{2} + x - 15 = 0$

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 $a = 5$ , $b = 1$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (5 \cdot - 15)}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 15}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4 \cdot 5 \cdot - 15$ .

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Step 5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 15}}{2 \cdot 5}$

Step 5.1.2.2

Multiply $- 20$ by $- 15$ .

$x = \frac{- 1 \pm \sqrt{1 + 300}}{2 \cdot 5}$

Step 5.1.3

Add $1$ and $300$ .

$x = \frac{- 1 \pm \sqrt{301}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{301}}{10}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 15}}{2 \cdot 5}$

Step 6.1.2

Multiply $- 4 \cdot 5 \cdot - 15$ .

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Step 6.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 15}}{2 \cdot 5}$

Step 6.1.2.2

Multiply $- 20$ by $- 15$ .

$x = \frac{- 1 \pm \sqrt{1 + 300}}{2 \cdot 5}$

Step 6.1.3

Add $1$ and $300$ .

$x = \frac{- 1 \pm \sqrt{301}}{2 \cdot 5}$

Step 6.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{301}}{10}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{301}}{10}$

Step 6.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{301}}{10}$

Step 6.5

Factor $- 1$ out of $\sqrt{301}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{301})}{10}$

Step 6.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{301})$ .

$x = \frac{- 1 (1 - \sqrt{301})}{10}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{301}}{10}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 15}}{2 \cdot 5}$

Step 7.1.2

Multiply $- 4 \cdot 5 \cdot - 15$ .

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Step 7.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 15}}{2 \cdot 5}$

Step 7.1.2.2

Multiply $- 20$ by $- 15$ .

$x = \frac{- 1 \pm \sqrt{1 + 300}}{2 \cdot 5}$

Step 7.1.3

Add $1$ and $300$ .

$x = \frac{- 1 \pm \sqrt{301}}{2 \cdot 5}$

Step 7.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{301}}{10}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{301}}{10}$

Step 7.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{301}}{10}$

Step 7.5

Factor $- 1$ out of $- \sqrt{301}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{301})}{10}$

Step 7.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{301})$ .

$x = \frac{- 1 (1 + \sqrt{301})}{10}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{301}}{10}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{301}}{10}, - \frac{1 + \sqrt{301}}{10}$

Step 9

The given equation $x = - \frac{1 - \sqrt{301}}{10}, - \frac{1 + \sqrt{301}}{10}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$