Finite Math Examples

$4 t^{2} - 60 t = 200$

Step 1

Subtract $200$ from both sides of the equation.

$4 t^{2} - 60 t - 200 = 0$

Step 2

Factor $4$ out of $4 t^{2} - 60 t - 200$ .

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Step 2.1

Factor $4$ out of $4 t^{2}$ .

$4 (t^{2}) - 60 t - 200 = 0$

Step 2.2

Factor $4$ out of $- 60 t$ .

$4 (t^{2}) + 4 (- 15 t) - 200 = 0$

Step 2.3

Factor $4$ out of $- 200$ .

$4 t^{2} + 4 (- 15 t) + 4 \cdot - 50 = 0$

Step 2.4

Factor $4$ out of $4 t^{2} + 4 (- 15 t)$ .

$4 (t^{2} - 15 t) + 4 \cdot - 50 = 0$

Step 2.5

Factor $4$ out of $4 (t^{2} - 15 t) + 4 \cdot - 50$ .

$4 (t^{2} - 15 t - 50) = 0$

Step 3

Divide each term in $4 (t^{2} - 15 t - 50) = 0$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 (t^{2} - 15 t - 50) = 0$ by $4$ .

$\frac{4 (t^{2} - 15 t - 50)}{4} = \frac{0}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 (t^{2} - 15 t - 50)}{4} = \frac{0}{4}$

Step 3.2.1.2

Divide $t^{2} - 15 t - 50$ by $1$ .

$t^{2} - 15 t - 50 = \frac{0}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $4$ .

$t^{2} - 15 t - 50 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 15$ , and $c = - 50$ into the quadratic formula and solve for $t$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot - 50)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 15$ to the power of $2$ .

$t = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot - 50}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 50$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{15 \pm \sqrt{225 - 4 \cdot - 50}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 50$ .

$t = \frac{15 \pm \sqrt{225 + 200}}{2 \cdot 1}$

Step 6.1.3

Add $225$ and $200$ .

$t = \frac{15 \pm \sqrt{425}}{2 \cdot 1}$

Step 6.1.4

Rewrite $425$ as $5^{2} \cdot 17$ .

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Step 6.1.4.1

Factor $25$ out of $425$ .

$t = \frac{15 \pm \sqrt{25 (17)}}{2 \cdot 1}$

Step 6.1.4.2

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

$t = \frac{15 \pm \sqrt{5^{2} \cdot 17}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$t = \frac{15 \pm 5 \sqrt{17}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$t = \frac{15 \pm 5 \sqrt{17}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{15 \pm 5 \sqrt{17}}{2}$

Decimal Form:

$t = 17.80776406 \dots, - 2.80776406 \dots$

Step 8