Solve Using the Quadratic Formula (r-2)(r+12)=-4

$(r - 2) (r + 12) = - 4$

Step 1

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

Simplify the left side.

Simplify $(r - 2) (r + 12)$ .

Expand $(r - 2) (r + 12)$ using the FOIL Method.

Apply the distributive property.

$r (r + 12) - 2 (r + 12) = - 4$

Apply the distributive property.

$r \cdot r + r \cdot 12 - 2 (r + 12) = - 4$

Apply the distributive property.

$r \cdot r + r \cdot 12 - 2 r - 2 \cdot 12 = - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

$r^{2} + r \cdot 12 - 2 r - 2 \cdot 12 = - 4$

Move $12$ to the left of $r$ .

$r^{2} + 12 \cdot r - 2 r - 2 \cdot 12 = - 4$

Multiply $- 2$ by $12$ .

$r^{2} + 12 r - 2 r - 24 = - 4$

Subtract $2 r$ from $12 r$ .

$r^{2} + 10 r - 24 = - 4$

Add $4$ to both sides of the equation.

$r^{2} + 10 r - 24 + 4 = 0$

Add $- 24$ and $4$ .

$r^{2} + 10 r - 20 = 0$

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 $a = 1$ , $b = 10$ , and $c = - 20$ into the quadratic formula and solve for $r$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$r = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 20$ .

Multiply $- 4$ by $1$ .

$r = \frac{- 10 \pm \sqrt{100 - 4 \cdot - 20}}{2 \cdot 1}$

Multiply $- 4$ by $- 20$ .

$r = \frac{- 10 \pm \sqrt{100 + 80}}{2 \cdot 1}$

Add $100$ and $80$ .

$r = \frac{- 10 \pm \sqrt{180}}{2 \cdot 1}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$r = \frac{- 10 \pm \sqrt{36 (5)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$r = \frac{- 10 \pm \sqrt{6^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{- 10 \pm 6 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{- 10 \pm 6 \sqrt{5}}{2}$

Simplify $\frac{- 10 \pm 6 \sqrt{5}}{2}$ .

$r = - 5 \pm 3 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = - 5 \pm 3 \sqrt{5}$

Decimal Form:

$r = 1.70820393 \dots, - 11.70820393 \dots$

Solve Using the Quadratic Formula (r-2)(r+12)=-4

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