Solve Using the Quadratic Formula 25=45t-5t^2

$25 = 45 t - 5 t^{2}$

Step 1

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

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

$25 - 45 t = - 5 t^{2}$

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

$25 - 45 t + 5 t^{2} = 0$

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 $c = 25$ into the quadratic formula and solve for $t$ .

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

Step 4

Simplify.

Simplify the numerator.

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

$t = \frac{45 \pm \sqrt{2025 - 4 \cdot 5 \cdot 25}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 25$ .

Multiply $- 4$ by $5$ .

$t = \frac{45 \pm \sqrt{2025 - 20 \cdot 25}}{2 \cdot 5}$

Multiply $- 20$ by $25$ .

$t = \frac{45 \pm \sqrt{2025 - 500}}{2 \cdot 5}$

Subtract $500$ from $2025$ .

$t = \frac{45 \pm \sqrt{1525}}{2 \cdot 5}$

Rewrite $1525$ as $5^{2} \cdot 61$ .

Factor $25$ out of $1525$ .

$t = \frac{45 \pm \sqrt{25 (61)}}{2 \cdot 5}$

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

$t = \frac{45 \pm \sqrt{5^{2} \cdot 61}}{2 \cdot 5}$

Pull terms out from under the radical.

$t = \frac{45 \pm 5 \sqrt{61}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$t = \frac{45 \pm 5 \sqrt{61}}{10}$

Simplify $\frac{45 \pm 5 \sqrt{61}}{10}$ .

$t = \frac{9 \pm \sqrt{61}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{9 \pm \sqrt{61}}{2}$

Decimal Form:

$t = 8.40512483 \dots, 0.59487516 \dots$