Basic Math Examples

$h (t) = - 4.9 t^{2} + 5 t + 9$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 4.9 t^{2} + 5 t + 9 = h t$

Step 2

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

$- 4.9 t^{2} + 5 t + 9 - h t = 0$

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 $a = - 4.9$ , $b = 5 - h$ , and $c = 9$ into the quadratic formula and solve for $t$ .

$\frac{- (5 - h) \pm \sqrt{{(5 - h)}^{2} - 4 \cdot (- 4.9 \cdot 9)}}{2 \cdot - 4.9}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$t = \frac{- 1 \cdot 5 + h \pm \sqrt{{(5 - h)}^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.2

Multiply $- 1$ by $5$ .

$t = \frac{- 5 + h \pm \sqrt{{(5 - h)}^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.3

Multiply $- - h$ .

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

$t = \frac{- 5 + 1 h \pm \sqrt{{(5 - h)}^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.3.2

Multiply $h$ by $1$ .

$t = \frac{- 5 + h \pm \sqrt{{(5 - h)}^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.4

Rewrite ${(5 - h)}^{2}$ as $(5 - h) (5 - h)$ .

$t = \frac{- 5 + h \pm \sqrt{(5 - h) (5 - h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.5

Expand $(5 - h) (5 - h)$ using the FOIL Method.

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Step 5.1.5.1

Apply the distributive property.

$t = \frac{- 5 + h \pm \sqrt{5 (5 - h) - h (5 - h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.5.2

Apply the distributive property.

$t = \frac{- 5 + h \pm \sqrt{5 \cdot 5 + 5 (- h) - h (5 - h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.5.3

Apply the distributive property.

$t = \frac{- 5 + h \pm \sqrt{5 \cdot 5 + 5 (- h) - h \cdot 5 - h (- h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6

Simplify and combine like terms.

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Step 5.1.6.1

Simplify each term.

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Step 5.1.6.1.1

Multiply $5$ by $5$ .

$t = \frac{- 5 + h \pm \sqrt{25 + 5 (- h) - h \cdot 5 - h (- h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.2

Multiply $- 1$ by $5$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - h \cdot 5 - h (- h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.3

Multiply $5$ by $- 1$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - 5 h - h (- h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.4

Rewrite using the commutative property of multiplication.

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - 5 h - 1 \cdot (- 1 h \cdot h) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.5

Multiply $h$ by $h$ by adding the exponents.

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Step 5.1.6.1.5.1

Move $h$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - 5 h - 1 \cdot (- 1 (h \cdot h)) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.5.2

Multiply $h$ by $h$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - 5 h - 1 \cdot (- 1 h^{2}) - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - 5 h + 1 h^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.1.7

Multiply $h^{2}$ by $1$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 5 h - 5 h + h^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.6.2

Subtract $5 h$ from $- 5 h$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 10 h + h^{2} - 4 \cdot - 4.9 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.7

Multiply $- 4 \cdot - 4.9 \cdot 9$ .

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Step 5.1.7.1

Multiply $- 4$ by $- 4.9$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 10 h + h^{2} + 19.6 \cdot 9}}{2 \cdot - 4.9}$

Step 5.1.7.2

Multiply $19.6$ by $9$ .

$t = \frac{- 5 + h \pm \sqrt{25 - 10 h + h^{2} + 176.4}}{2 \cdot - 4.9}$

Step 5.1.8

Add $25$ and $176.4$ .

$t = \frac{- 5 + h \pm \sqrt{- 10 h + h^{2} + 201.4}}{2 \cdot - 4.9}$

Step 5.1.9

Reorder terms.

$t = \frac{- 5 + h \pm \sqrt{h^{2} - 10 h + 201.4}}{2 \cdot - 4.9}$

Step 5.2

Multiply $2$ by $- 4.9$ .

$t = \frac{- 5 + h \pm \sqrt{h^{2} - 10 h + 201.4}}{- 9.8}$

Step 5.3

Simplify $\frac{- 5 + h \pm \sqrt{h^{2} - 10 h + 201.4}}{- 9.8}$ .

$t = \frac{25 - 5 h \pm 5 \sqrt{h^{2} - 10 h + 201.4}}{49}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{5 (5 - h + \sqrt{h^{2} - 10 h + 201.4})}{49}$

$t = \frac{5 (5 - h - \sqrt{h^{2} - 10 h + 201.4})}{49}$