Solve for r r(r+5)(2)=90

$r (r + 5) (2) = 90$

Divide each term by $2$ and simplify.

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Divide each term in $r (r + 5) (2) = 90$ by $2$ .

$\frac{r (r + 5) (2)}{2} = \frac{90}{2}$

Simplify $\frac{r (r + 5) (2)}{2}$ .

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{r (r + 5) \cdot 2}{2} = \frac{90}{2}$

Divide $r (r + 5)$ by $1$ .

$r (r + 5) = \frac{90}{2}$

Apply the distributive property.

$r \cdot r + r \cdot 5 = \frac{90}{2}$

Simplify the expression.

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Multiply $r$ by $r$ .

$r^{2} + r \cdot 5 = \frac{90}{2}$

Move $5$ to the left of $r$ .

$r^{2} + 5 r = \frac{90}{2}$

Divide $90$ by $2$ .

$r^{2} + 5 r = 45$

Move $45$ to the left side of the equation by subtracting it from both sides.

$r^{2} + 5 r - 45 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 5$ , and $c = - 45$ into the quadratic formula and solve for $r$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 45)}}{2 \cdot 1}$

Simplify.

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Simplify the numerator.

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Raise $5$ to the power of $2$ .

$r = \frac{- 5 \pm \sqrt{25 - 4 \cdot (1 \cdot - 45)}}{2 \cdot 1}$

Multiply $- 45$ by $1$ .

$r = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 45}}{2 \cdot 1}$

Multiply $- 4$ by $- 45$ .

$r = \frac{- 5 \pm \sqrt{25 + 180}}{2 \cdot 1}$

Add $25$ and $180$ .

$r = \frac{- 5 \pm \sqrt{205}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{- 5 \pm \sqrt{205}}{2}$

The final answer is the combination of both solutions.

$r = - \frac{5 - \sqrt{205}}{2}, - \frac{5 + \sqrt{205}}{2}$

The result can be shown in multiple forms.

Exact Form:

$r = - \frac{5 - \sqrt{205}}{2}, - \frac{5 + \sqrt{205}}{2}$

Decimal Form:

$r = 4.65891053 \dots, - 9.65891053 \dots$

Solve for r r(r+5)(2)=90

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