Solve Using the Quadratic Formula (r-3)(r+7)=-13

$(r - 3) (r + 7) = - 13$

Step 1

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

Simplify the left side.

Simplify $(r - 3) (r + 7)$ .

Expand $(r - 3) (r + 7)$ using the FOIL Method.

Apply the distributive property.

$r (r + 7) - 3 (r + 7) = - 13$

Apply the distributive property.

$r \cdot r + r \cdot 7 - 3 (r + 7) = - 13$

Apply the distributive property.

$r \cdot r + r \cdot 7 - 3 r - 3 \cdot 7 = - 13$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

$r^{2} + r \cdot 7 - 3 r - 3 \cdot 7 = - 13$

Move $7$ to the left of $r$ .

$r^{2} + 7 \cdot r - 3 r - 3 \cdot 7 = - 13$

Multiply $- 3$ by $7$ .

$r^{2} + 7 r - 3 r - 21 = - 13$

Subtract $3 r$ from $7 r$ .

$r^{2} + 4 r - 21 = - 13$

Add $13$ to both sides of the equation.

$r^{2} + 4 r - 21 + 13 = 0$

Add $- 21$ and $13$ .

$r^{2} + 4 r - 8 = 0$

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 $a = 1$ , $b = 4$ , and $c = - 8$ into the quadratic formula and solve for $r$ .

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

Step 4

Simplify.

Simplify the numerator.

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

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 8$ .

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Multiply $- 4$ by $- 8$ .

$r = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Add $16$ and $32$ .

$r = \frac{- 4 \pm \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$r = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

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

$r = \frac{- 4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{- 4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 4 \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{2}$ .

$r = - 2 \pm 2 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = - 2 \pm 2 \sqrt{3}$

Decimal Form:

$r = 1.46410161 \dots, - 5.46410161 \dots$

Solve Using the Quadratic Formula (r-3)(r+7)=-13

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