Precalculus Examples

$15 x^{2} + 58 x + 48 = 0$

Step 1

Subtract $48$ from both sides of the equation.

$15 x^{2} + 58 x = - 48$

Step 2

Divide each term in $15 x^{2} + 58 x = - 48$ by $15$ and simplify.

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Step 2.1

Divide each term in $15 x^{2} + 58 x = - 48$ by $15$ .

$\frac{15 x^{2}}{15} + \frac{58 x}{15} = \frac{- 48}{15}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $15$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{15 x^{2}}{15} + \frac{58 x}{15} = \frac{- 48}{15}$

Step 2.2.1.2

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

$x^{2} + \frac{58 x}{15} = \frac{- 48}{15}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $- 48$ and $15$ .

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Step 2.3.1.1

Factor $3$ out of $- 48$ .

$x^{2} + \frac{58 x}{15} = \frac{3 (- 16)}{15}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $3$ out of $15$ .

$x^{2} + \frac{58 x}{15} = \frac{3 \cdot - 16}{3 \cdot 5}$

Step 2.3.1.2.2

Cancel the common factor.

$x^{2} + \frac{58 x}{15} = \frac{3 \cdot - 16}{3 \cdot 5}$

Step 2.3.1.2.3

Rewrite the expression.

$x^{2} + \frac{58 x}{15} = \frac{- 16}{5}$

Step 2.3.2

Move the negative in front of the fraction.

$x^{2} + \frac{58 x}{15} = - \frac{16}{5}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{29}{15})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{58 x}{15} + {(\frac{29}{15})}^{2} = - \frac{16}{5} + {(\frac{29}{15})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $\frac{29}{15}$ .

$x^{2} + \frac{58 x}{15} + \frac{29^{2}}{15^{2}} = - \frac{16}{5} + {(\frac{29}{15})}^{2}$

Step 5.1.1.2

Raise $29$ to the power of $2$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{15^{2}} = - \frac{16}{5} + {(\frac{29}{15})}^{2}$

Step 5.1.1.3

Raise $15$ to the power of $2$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16}{5} + {(\frac{29}{15})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{16}{5} + {(\frac{29}{15})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the product rule to $\frac{29}{15}$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16}{5} + \frac{29^{2}}{15^{2}}$

Step 5.2.1.1.2

Raise $29$ to the power of $2$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16}{5} + \frac{841}{15^{2}}$

Step 5.2.1.1.3

Raise $15$ to the power of $2$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16}{5} + \frac{841}{225}$

Step 5.2.1.2

To write $- \frac{16}{5}$ as a fraction with a common denominator, multiply by $\frac{45}{45}$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16}{5} \cdot \frac{45}{45} + \frac{841}{225}$

Step 5.2.1.3

Write each expression with a common denominator of $225$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{16}{5}$ by $\frac{45}{45}$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16 \cdot 45}{5 \cdot 45} + \frac{841}{225}$

Step 5.2.1.3.2

Multiply $5$ by $45$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = - \frac{16 \cdot 45}{225} + \frac{841}{225}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = \frac{- 16 \cdot 45 + 841}{225}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $- 16$ by $45$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = \frac{- 720 + 841}{225}$

Step 5.2.1.5.2

Add $- 720$ and $841$ .

$x^{2} + \frac{58 x}{15} + \frac{841}{225} = \frac{121}{225}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{29}{15})}^{2}$ .

${(x + \frac{29}{15})}^{2} = \frac{121}{225}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{29}{15} = \pm \sqrt{\frac{121}{225}}$

Step 7.2

Simplify $\pm \sqrt{\frac{121}{225}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{121}{225}}$ as $\frac{\sqrt{121}}{\sqrt{225}}$ .

$x + \frac{29}{15} = \pm \frac{\sqrt{121}}{\sqrt{225}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $121$ as $11^{2}$ .

$x + \frac{29}{15} = \pm \frac{\sqrt{11^{2}}}{\sqrt{225}}$

Step 7.2.2.2

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

$x + \frac{29}{15} = \pm \frac{11}{\sqrt{225}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $225$ as $15^{2}$ .

$x + \frac{29}{15} = \pm \frac{11}{\sqrt{15^{2}}}$

Step 7.2.3.2

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

$x + \frac{29}{15} = \pm \frac{11}{15}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x + \frac{29}{15} = \frac{11}{15}$

Step 7.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.2.1

Subtract $\frac{29}{15}$ from both sides of the equation.

$x = \frac{11}{15} - \frac{29}{15}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$x = \frac{11 - 29}{15}$

Step 7.3.2.3

Subtract $29$ from $11$ .

$x = \frac{- 18}{15}$

Step 7.3.2.4

Cancel the common factor of $- 18$ and $15$ .

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Step 7.3.2.4.1

Factor $3$ out of $- 18$ .

$x = \frac{3 (- 6)}{15}$

Step 7.3.2.4.2

Cancel the common factors.

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Step 7.3.2.4.2.1

Factor $3$ out of $15$ .

$x = \frac{3 \cdot - 6}{3 \cdot 5}$

Step 7.3.2.4.2.2

Cancel the common factor.

$x = \frac{3 \cdot - 6}{3 \cdot 5}$

Step 7.3.2.4.2.3

Rewrite the expression.

$x = \frac{- 6}{5}$

Step 7.3.2.5

Move the negative in front of the fraction.

$x = - \frac{6}{5}$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + \frac{29}{15} = - \frac{11}{15}$

Step 7.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.4.1

Subtract $\frac{29}{15}$ from both sides of the equation.

$x = - \frac{11}{15} - \frac{29}{15}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$x = \frac{- 11 - 29}{15}$

Step 7.3.4.3

Subtract $29$ from $- 11$ .

$x = \frac{- 40}{15}$

Step 7.3.4.4

Cancel the common factor of $- 40$ and $15$ .

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Step 7.3.4.4.1

Factor $5$ out of $- 40$ .

$x = \frac{5 (- 8)}{15}$

Step 7.3.4.4.2

Cancel the common factors.

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Step 7.3.4.4.2.1

Factor $5$ out of $15$ .

$x = \frac{5 \cdot - 8}{5 \cdot 3}$

Step 7.3.4.4.2.2

Cancel the common factor.

$x = \frac{5 \cdot - 8}{5 \cdot 3}$

Step 7.3.4.4.2.3

Rewrite the expression.

$x = \frac{- 8}{3}$

Step 7.3.4.5

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{6}{5}, - \frac{8}{3}$