Finite Math Examples

$(r - 3) (r + 17) = - 20$

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 $(r - 3) (r + 17)$ .

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Step 1.1.1.1

Expand $(r - 3) (r + 17)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$r (r + 17) - 3 (r + 17) = - 20$

Step 1.1.1.1.2

Apply the distributive property.

$r \cdot r + r \cdot 17 - 3 (r + 17) = - 20$

Step 1.1.1.1.3

Apply the distributive property.

$r \cdot r + r \cdot 17 - 3 r - 3 \cdot 17 = - 20$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Multiply $r$ by $r$ .

$r^{2} + r \cdot 17 - 3 r - 3 \cdot 17 = - 20$

Step 1.1.1.2.1.2

Move $17$ to the left of $r$ .

$r^{2} + 17 \cdot r - 3 r - 3 \cdot 17 = - 20$

Step 1.1.1.2.1.3

Multiply $- 3$ by $17$ .

$r^{2} + 17 r - 3 r - 51 = - 20$

Step 1.1.1.2.2

Subtract $3 r$ from $17 r$ .

$r^{2} + 14 r - 51 = - 20$

Step 1.2

Add $20$ to both sides of the equation.

$r^{2} + 14 r - 51 + 20 = 0$

Step 1.3

Add $- 51$ and $20$ .

$r^{2} + 14 r - 31 = 0$

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 $a = 1$ , $b = 14$ , and $c = - 31$ into the quadratic formula and solve for $r$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (1 \cdot - 31)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $14$ to the power of $2$ .

$r = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot - 31}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 31$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 14 \pm \sqrt{196 - 4 \cdot - 31}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 31$ .

$r = \frac{- 14 \pm \sqrt{196 + 124}}{2 \cdot 1}$

Step 4.1.3

Add $196$ and $124$ .

$r = \frac{- 14 \pm \sqrt{320}}{2 \cdot 1}$

Step 4.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 4.1.4.1

Factor $64$ out of $320$ .

$r = \frac{- 14 \pm \sqrt{64 (5)}}{2 \cdot 1}$

Step 4.1.4.2

Rewrite $64$ as $8^{2}$ .

$r = \frac{- 14 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$r = \frac{- 14 \pm 8 \sqrt{5}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$r = \frac{- 14 \pm 8 \sqrt{5}}{2}$

Step 4.3

Simplify $\frac{- 14 \pm 8 \sqrt{5}}{2}$ .

$r = - 7 \pm 4 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = - 7 \pm 4 \sqrt{5}$

Decimal Form:

$r = 1.94427190 \dots, - 15.94427190 \dots$