Solve Using the Quadratic Formula 27=60t-5t^2

$27 = 60 t - 5 t^{2}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $60 t$ from both sides of the equation.

$27 - 60 t = - 5 t^{2}$

Add $5 t^{2}$ to both sides of the equation.

$27 - 60 t + 5 t^{2} = 0$

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 $a = 5$ , $b = - 60$ , and $c = 27$ into the quadratic formula and solve for $t$ .

$\frac{60 \pm \sqrt{{(- 60)}^{2} - 4 \cdot (5 \cdot 27)}}{2 \cdot 5}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 60$ to the power of $2$ .

$t = \frac{60 \pm \sqrt{3600 - 4 \cdot 5 \cdot 27}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 27$ .

Multiply $- 4$ by $5$ .

$t = \frac{60 \pm \sqrt{3600 - 20 \cdot 27}}{2 \cdot 5}$

Multiply $- 20$ by $27$ .

$t = \frac{60 \pm \sqrt{3600 - 540}}{2 \cdot 5}$

Subtract $540$ from $3600$ .

$t = \frac{60 \pm \sqrt{3060}}{2 \cdot 5}$

Rewrite $3060$ as $6^{2} \cdot 85$ .

Factor $36$ out of $3060$ .

$t = \frac{60 \pm \sqrt{36 (85)}}{2 \cdot 5}$

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

$t = \frac{60 \pm \sqrt{6^{2} \cdot 85}}{2 \cdot 5}$

Pull terms out from under the radical.

$t = \frac{60 \pm 6 \sqrt{85}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$t = \frac{60 \pm 6 \sqrt{85}}{10}$

Simplify $\frac{60 \pm 6 \sqrt{85}}{10}$ .

$t = \frac{30 \pm 3 \sqrt{85}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{30 \pm 3 \sqrt{85}}{5}$

Decimal Form:

$t = 11.53172667 \dots, 0.46827332 \dots$