Solve Using the Square Root Property r^2-4r=30

$r^{2} - 4 r = 30$

Step 1

Subtract $30$ from both sides of the equation.

$r^{2} - 4 r - 30 = 0$

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 $c = - 30$ into the quadratic formula and solve for $r$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 30)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$r = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 30$ .

Multiply $- 4$ by $1$ .

$r = \frac{4 \pm \sqrt{16 - 4 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4$ by $- 30$ .

$r = \frac{4 \pm \sqrt{16 + 120}}{2 \cdot 1}$

Add $16$ and $120$ .

$r = \frac{4 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$r = \frac{4 \pm \sqrt{4 (34)}}{2 \cdot 1}$

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

$r = \frac{4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{4 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{4 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{34}}{2}$ .

$r = 2 \pm \sqrt{34}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

$r = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 30$ .

Multiply $- 4$ by $1$ .

$r = \frac{4 \pm \sqrt{16 - 4 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4$ by $- 30$ .

$r = \frac{4 \pm \sqrt{16 + 120}}{2 \cdot 1}$

Add $16$ and $120$ .

$r = \frac{4 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$r = \frac{4 \pm \sqrt{4 (34)}}{2 \cdot 1}$

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

$r = \frac{4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{4 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{4 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{34}}{2}$ .

$r = 2 \pm \sqrt{34}$

Change the $\pm$ to $+$ .

$r = 2 + \sqrt{34}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

$r = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 30$ .

Multiply $- 4$ by $1$ .

$r = \frac{4 \pm \sqrt{16 - 4 \cdot - 30}}{2 \cdot 1}$

Multiply $- 4$ by $- 30$ .

$r = \frac{4 \pm \sqrt{16 + 120}}{2 \cdot 1}$

Add $16$ and $120$ .

$r = \frac{4 \pm \sqrt{136}}{2 \cdot 1}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$r = \frac{4 \pm \sqrt{4 (34)}}{2 \cdot 1}$

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

$r = \frac{4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{4 \pm 2 \sqrt{34}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{4 \pm 2 \sqrt{34}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{34}}{2}$ .

$r = 2 \pm \sqrt{34}$

Change the $\pm$ to $-$ .

$r = 2 - \sqrt{34}$

Step 7

The final answer is the combination of both solutions.

$r = 2 + \sqrt{34}, 2 - \sqrt{34}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$r = 2 + \sqrt{34}, 2 - \sqrt{34}$

Decimal Form:

$r = 7.83095189 \dots, - 3.83095189 \dots$

Solve Using the Square Root Property r^2-4r=30

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