Solve Using the Quadratic Formula (r-4)(r+12)=-37

$(r - 4) (r + 12) = - 37$

Step 1

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

Simplify the left side.

Simplify $(r - 4) (r + 12)$ .

Expand $(r - 4) (r + 12)$ using the FOIL Method.

Apply the distributive property.

$r (r + 12) - 4 (r + 12) = - 37$

Apply the distributive property.

$r \cdot r + r \cdot 12 - 4 (r + 12) = - 37$

Apply the distributive property.

$r \cdot r + r \cdot 12 - 4 r - 4 \cdot 12 = - 37$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

$r^{2} + r \cdot 12 - 4 r - 4 \cdot 12 = - 37$

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

$r^{2} + 12 \cdot r - 4 r - 4 \cdot 12 = - 37$

Multiply $- 4$ by $12$ .

$r^{2} + 12 r - 4 r - 48 = - 37$

Subtract $4 r$ from $12 r$ .

$r^{2} + 8 r - 48 = - 37$

Add $37$ to both sides of the equation.

$r^{2} + 8 r - 48 + 37 = 0$

Add $- 48$ and $37$ .

$r^{2} + 8 r - 11 = 0$

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 $a = 1$ , $b = 8$ , and $c = - 11$ into the quadratic formula and solve for $r$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $8$ to the power of $2$ .

$r = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$r = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$r = \frac{- 8 \pm \sqrt{64 + 44}}{2 \cdot 1}$

Add $64$ and $44$ .

$r = \frac{- 8 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$r = \frac{- 8 \pm \sqrt{36 (3)}}{2 \cdot 1}$

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

$r = \frac{- 8 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{- 8 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{- 8 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{- 8 \pm 6 \sqrt{3}}{2}$ .

$r = - 4 \pm 3 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = - 4 \pm 3 \sqrt{3}$

Decimal Form:

$r = 1.19615242 \dots, - 9.19615242 \dots$

Solve Using the Quadratic Formula (r-4)(r+12)=-37

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