Solve Using the Quadratic Formula 20=45t-5t^2

$20 = 45 t - 5 t^{2}$

Step 1

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

Subtract $45 t$ from both sides of the equation.

$20 - 45 t = - 5 t^{2}$

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

$20 - 45 t + 5 t^{2} = 0$

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 $a = 5$ , $b = - 45$ , and $c = 20$ into the quadratic formula and solve for $t$ .

$\frac{45 \pm \sqrt{{(- 45)}^{2} - 4 \cdot (5 \cdot 20)}}{2 \cdot 5}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 45$ to the power of $2$ .

$t = \frac{45 \pm \sqrt{2025 - 4 \cdot 5 \cdot 20}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 20$ .

Multiply $- 4$ by $5$ .

$t = \frac{45 \pm \sqrt{2025 - 20 \cdot 20}}{2 \cdot 5}$

Multiply $- 20$ by $20$ .

$t = \frac{45 \pm \sqrt{2025 - 400}}{2 \cdot 5}$

Subtract $400$ from $2025$ .

$t = \frac{45 \pm \sqrt{1625}}{2 \cdot 5}$

Rewrite $1625$ as $5^{2} \cdot 65$ .

Factor $25$ out of $1625$ .

$t = \frac{45 \pm \sqrt{25 (65)}}{2 \cdot 5}$

Rewrite $25$ as $5^{2}$ .

$t = \frac{45 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 5}$

Pull terms out from under the radical.

$t = \frac{45 \pm 5 \sqrt{65}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$t = \frac{45 \pm 5 \sqrt{65}}{10}$

Simplify $\frac{45 \pm 5 \sqrt{65}}{10}$ .

$t = \frac{9 \pm \sqrt{65}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{9 \pm \sqrt{65}}{2}$

Decimal Form:

$t = 8.53112887 \dots, 0.46887112 \dots$