Finite Math Examples

$h = - 0.8 t^{2} + 6 t$

Step 1

Rewrite the equation as $- 0.8 t^{2} + 6 t = h$ .

$- 0.8 t^{2} + 6 t = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 0.8 t^{2} + 6 t - h = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 0.8$ , $b = 6$ , and $c = - h$ into the quadratic formula and solve for $t$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 0.8 \cdot (- h))}}{2 \cdot - 0.8}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $2$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 0.8 \cdot (- 1 h)}}{2 \cdot - 0.8}$

Step 5.1.2

Multiply $- 4 \cdot - 0.8 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 0.8$ .

$t = \frac{- 6 \pm \sqrt{36 + 3.2 \cdot (- 1 h)}}{2 \cdot - 0.8}$

Step 5.1.2.2

Multiply $3.2$ by $- 1$ .

$t = \frac{- 6 \pm \sqrt{36 - 3.2 h}}{2 \cdot - 0.8}$

Step 5.1.3

Factor $0.8$ out of $36 - 3.2 h$ .

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Step 5.1.3.1

Factor $0.8$ out of $36$ .

$t = \frac{- 6 \pm \sqrt{0.8 (45) - 3.2 h}}{2 \cdot - 0.8}$

Step 5.1.3.2

Factor $0.8$ out of $- 3.2 h$ .

$t = \frac{- 6 \pm \sqrt{0.8 (45) + 0.8 (- 4 h)}}{2 \cdot - 0.8}$

Step 5.1.3.3

Factor $0.8$ out of $0.8 (45) + 0.8 (- 4 h)$ .

$t = \frac{- 6 \pm \sqrt{0.8 (45 - 4 h)}}{2 \cdot - 0.8}$

Step 5.1.4

Rewrite $0.8 (45 - 4 h)$ as ${({0.9457416}^{2})}^{2} (45 - 4 h)$ .

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Step 5.1.4.1

Rewrite $0.8$ as ${0.89442719}^{2}$ .

$t = \frac{- 6 \pm \sqrt{{0.89442719}^{2} (45 - 4 h)}}{2 \cdot - 0.8}$

Step 5.1.4.2

Rewrite $0.89442719$ as ${0.9457416}^{2}$ .

$t = \frac{- 6 \pm \sqrt{{({0.9457416}^{2})}^{2} (45 - 4 h)}}{2 \cdot - 0.8}$

Step 5.1.5

Pull terms out from under the radical.

$t = \frac{- 6 \pm {0.9457416}^{2} \sqrt{45 - 4 h}}{2 \cdot - 0.8}$

Step 5.1.6

Raise $0.9457416$ to the power of $2$ .

$t = \frac{- 6 \pm 0.89442719 \sqrt{45 - 4 h}}{2 \cdot - 0.8}$

Step 5.2

Multiply $2$ by $- 0.8$ .

$t = \frac{- 6 \pm 0.89442719 \sqrt{45 - 4 h}}{- 1.6}$

Step 5.3

Simplify $\frac{- 6 \pm 0.89442719 \sqrt{45 - 4 h}}{- 1.6}$ .

$t = \frac{30 \pm 4.47213595 \sqrt{45 - 4 h}}{8}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{30 + 4.47213595 \sqrt{45 - 4 h}}{8}$

$t = \frac{30 - 4.47213595 \sqrt{45 - 4 h}}{8}$