Precalculus Examples

$2 r^{2} + 3 r = 9$

Step 1

Divide each term in $2 r^{2} + 3 r = 9$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 r^{2} + 3 r = 9$ by $2$ .

$\frac{2 r^{2}}{2} + \frac{3 r}{2} = \frac{9}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 r^{2}}{2} + \frac{3 r}{2} = \frac{9}{2}$

Step 1.2.1.2

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

$r^{2} + \frac{3 r}{2} = \frac{9}{2}$

Step 2

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

${(\frac{b}{2})}^{2} = {(\frac{3}{4})}^{2}$

Step 3

Add the term to each side of the equation.

$r^{2} + \frac{3 r}{2} + {(\frac{3}{4})}^{2} = \frac{9}{2} + {(\frac{3}{4})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\frac{3}{4}$ .

$r^{2} + \frac{3 r}{2} + \frac{3^{2}}{4^{2}} = \frac{9}{2} + {(\frac{3}{4})}^{2}$

Step 4.1.1.2

Raise $3$ to the power of $2$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{4^{2}} = \frac{9}{2} + {(\frac{3}{4})}^{2}$

Step 4.1.1.3

Raise $4$ to the power of $2$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9}{2} + {(\frac{3}{4})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{9}{2} + {(\frac{3}{4})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $\frac{3}{4}$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9}{2} + \frac{3^{2}}{4^{2}}$

Step 4.2.1.1.2

Raise $3$ to the power of $2$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9}{2} + \frac{9}{4^{2}}$

Step 4.2.1.1.3

Raise $4$ to the power of $2$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9}{2} + \frac{9}{16}$

Step 4.2.1.2

To write $\frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9}{2} \cdot \frac{8}{8} + \frac{9}{16}$

Step 4.2.1.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{9}{2}$ by $\frac{8}{8}$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9 \cdot 8}{2 \cdot 8} + \frac{9}{16}$

Step 4.2.1.3.2

Multiply $2$ by $8$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9 \cdot 8}{16} + \frac{9}{16}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{9 \cdot 8 + 9}{16}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $9$ by $8$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{72 + 9}{16}$

Step 4.2.1.5.2

Add $72$ and $9$ .

$r^{2} + \frac{3 r}{2} + \frac{9}{16} = \frac{81}{16}$

Step 5

Factor the perfect trinomial square into ${(r + \frac{3}{4})}^{2}$ .

${(r + \frac{3}{4})}^{2} = \frac{81}{16}$

Step 6

Solve the equation for $r$ .

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Step 6.1

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

$r + \frac{3}{4} = \pm \sqrt{\frac{81}{16}}$

Step 6.2

Simplify $\pm \sqrt{\frac{81}{16}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{81}{16}}$ as $\frac{\sqrt{81}}{\sqrt{16}}$ .

$r + \frac{3}{4} = \pm \frac{\sqrt{81}}{\sqrt{16}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Rewrite $81$ as $9^{2}$ .

$r + \frac{3}{4} = \pm \frac{\sqrt{9^{2}}}{\sqrt{16}}$

Step 6.2.2.2

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

$r + \frac{3}{4} = \pm \frac{9}{\sqrt{16}}$

Step 6.2.3

Simplify the denominator.

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Step 6.2.3.1

Rewrite $16$ as $4^{2}$ .

$r + \frac{3}{4} = \pm \frac{9}{\sqrt{4^{2}}}$

Step 6.2.3.2

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

$r + \frac{3}{4} = \pm \frac{9}{4}$

Step 6.3

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

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Step 6.3.1

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

$r + \frac{3}{4} = \frac{9}{4}$

Step 6.3.2

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

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Step 6.3.2.1

Subtract $\frac{3}{4}$ from both sides of the equation.

$r = \frac{9}{4} - \frac{3}{4}$

Step 6.3.2.2

Combine the numerators over the common denominator.

$r = \frac{9 - 3}{4}$

Step 6.3.2.3

Subtract $3$ from $9$ .

$r = \frac{6}{4}$

Step 6.3.2.4

Cancel the common factor of $6$ and $4$ .

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Step 6.3.2.4.1

Factor $2$ out of $6$ .

$r = \frac{2 (3)}{4}$

Step 6.3.2.4.2

Cancel the common factors.

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Step 6.3.2.4.2.1

Factor $2$ out of $4$ .

$r = \frac{2 \cdot 3}{2 \cdot 2}$

Step 6.3.2.4.2.2

Cancel the common factor.

$r = \frac{2 \cdot 3}{2 \cdot 2}$

Step 6.3.2.4.2.3

Rewrite the expression.

$r = \frac{3}{2}$

Step 6.3.3

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

$r + \frac{3}{4} = - \frac{9}{4}$

Step 6.3.4

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

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Step 6.3.4.1

Subtract $\frac{3}{4}$ from both sides of the equation.

$r = - \frac{9}{4} - \frac{3}{4}$

Step 6.3.4.2

Combine the numerators over the common denominator.

$r = \frac{- 9 - 3}{4}$

Step 6.3.4.3

Subtract $3$ from $- 9$ .

$r = \frac{- 12}{4}$

Step 6.3.4.4

Divide $- 12$ by $4$ .

$r = - 3$

Step 6.3.5

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

$r = \frac{3}{2}, - 3$