Finite Math Examples

$r (r + 11) = 80$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $r (r + 11)$ .

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Step 1.1.1.1

Apply the distributive property.

$r \cdot r + r \cdot 11 = 80$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Multiply $r$ by $r$ .

$r^{2} + r \cdot 11 = 80$

Step 1.1.1.2.2

Move $11$ to the left of $r$ .

$r^{2} + 11 r = 80$

Step 1.2

Subtract $80$ from both sides of the equation.

$r^{2} + 11 r - 80 = 0$

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 $a = 1$ , $b = 11$ , and $c = - 80$ into the quadratic formula and solve for $r$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (1 \cdot - 80)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $11$ to the power of $2$ .

$r = \frac{- 11 \pm \sqrt{121 - 4 \cdot 1 \cdot - 80}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 80$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 11 \pm \sqrt{121 - 4 \cdot - 80}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 80$ .

$r = \frac{- 11 \pm \sqrt{121 + 320}}{2 \cdot 1}$

Step 4.1.3

Add $121$ and $320$ .

$r = \frac{- 11 \pm \sqrt{441}}{2 \cdot 1}$

Step 4.1.4

Rewrite $441$ as $21^{2}$ .

$r = \frac{- 11 \pm \sqrt{21^{2}}}{2 \cdot 1}$

Step 4.1.5

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

$r = \frac{- 11 \pm 21}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$r = \frac{- 11 \pm 21}{2}$

Step 5

The final answer is the combination of both solutions.

$r = 5, - 16$