Pre-Algebra Examples

$- 16 t^{2} + 92 t + 24 = h (t)$

Step 1

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

$- 16 t^{2} + 92 t + 24 - h t = 0$

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 $a = - 16$ , $b = 92 - h$ , and $c = 24$ into the quadratic formula and solve for $t$ .

$\frac{- (92 - h) \pm \sqrt{{(92 - h)}^{2} - 4 \cdot (- 16 \cdot 24)}}{2 \cdot - 16}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$t = \frac{- 1 \cdot 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.2

Multiply $- 1$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.3

Multiply $- - h$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$t = \frac{- 92 + 1 h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.3.2

Multiply $h$ by $1$ .

$t = \frac{- 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.4

Rewrite ${(92 - h)}^{2}$ as $(92 - h) (92 - h)$ .

$t = \frac{- 92 + h \pm \sqrt{(92 - h) (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.5

Expand $(92 - h) (92 - h)$ using the FOIL Method.

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Step 4.1.5.1

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 (92 - h) - h (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.5.2

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 \cdot 92 + 92 (- h) - h (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.5.3

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 \cdot 92 + 92 (- h) - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6

Simplify and combine like terms.

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Step 4.1.6.1

Simplify each term.

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Step 4.1.6.1.1

Multiply $92$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{8464 + 92 (- h) - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.2

Multiply $- 1$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.3

Multiply $92$ by $- 1$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.4

Rewrite using the commutative property of multiplication.

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 h \cdot h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.5

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

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Step 4.1.6.1.5.1

Move $h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 (h \cdot h)) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.5.2

Multiply $h$ by $h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 h^{2}) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.6

Multiply $- 1$ by $- 1$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h + 1 h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.1.7

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

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h + h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.6.2

Subtract $92 h$ from $- 92 h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 4.1.7

Multiply $- 4 \cdot - 16 \cdot 24$ .

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Step 4.1.7.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} + 64 \cdot 24}}{2 \cdot - 16}$

Step 4.1.7.2

Multiply $64$ by $24$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} + 1536}}{2 \cdot - 16}$

Step 4.1.8

Add $8464$ and $1536$ .

$t = \frac{- 92 + h \pm \sqrt{- 184 h + h^{2} + 10000}}{2 \cdot - 16}$

Step 4.1.9

Reorder terms.

$t = \frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{2 \cdot - 16}$

Step 4.2

Multiply $2$ by $- 16$ .

$t = \frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{- 32}$

Step 4.3

Simplify $\frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{- 32}$ .

$t = \frac{92 - h \pm \sqrt{h^{2} - 184 h + 10000}}{32}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$t = \frac{- 1 \cdot 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 1$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.3

Multiply $- - h$ .

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

$t = \frac{- 92 + 1 h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.3.2

Multiply $h$ by $1$ .

$t = \frac{- 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.4

Rewrite ${(92 - h)}^{2}$ as $(92 - h) (92 - h)$ .

$t = \frac{- 92 + h \pm \sqrt{(92 - h) (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.5

Expand $(92 - h) (92 - h)$ using the FOIL Method.

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Step 5.1.5.1

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 (92 - h) - h (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.5.2

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 \cdot 92 + 92 (- h) - h (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.5.3

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 \cdot 92 + 92 (- h) - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

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 $92$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{8464 + 92 (- h) - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.1.2

Multiply $- 1$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.1.3

Multiply $92$ by $- 1$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.1.4

Rewrite using the commutative property of multiplication.

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 h \cdot h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

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 $h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 (h \cdot h)) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.1.5.2

Multiply $h$ by $h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 h^{2}) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h + 1 h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.1.7

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

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h + h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.6.2

Subtract $92 h$ from $- 92 h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 5.1.7

Multiply $- 4 \cdot - 16 \cdot 24$ .

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Step 5.1.7.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} + 64 \cdot 24}}{2 \cdot - 16}$

Step 5.1.7.2

Multiply $64$ by $24$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} + 1536}}{2 \cdot - 16}$

Step 5.1.8

Add $8464$ and $1536$ .

$t = \frac{- 92 + h \pm \sqrt{- 184 h + h^{2} + 10000}}{2 \cdot - 16}$

Step 5.1.9

Reorder terms.

$t = \frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{- 32}$

Step 5.3

Simplify $\frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{- 32}$ .

$t = \frac{92 - h \pm \sqrt{h^{2} - 184 h + 10000}}{32}$

Step 5.4

Change the $\pm$ to $+$ .

$t = \frac{92 - h + \sqrt{h^{2} - 184 h + 10000}}{32}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$t = \frac{- 1 \cdot 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.2

Multiply $- 1$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.3

Multiply $- - h$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$t = \frac{- 92 + 1 h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.3.2

Multiply $h$ by $1$ .

$t = \frac{- 92 + h \pm \sqrt{{(92 - h)}^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.4

Rewrite ${(92 - h)}^{2}$ as $(92 - h) (92 - h)$ .

$t = \frac{- 92 + h \pm \sqrt{(92 - h) (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.5

Expand $(92 - h) (92 - h)$ using the FOIL Method.

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Step 6.1.5.1

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 (92 - h) - h (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.5.2

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 \cdot 92 + 92 (- h) - h (92 - h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.5.3

Apply the distributive property.

$t = \frac{- 92 + h \pm \sqrt{92 \cdot 92 + 92 (- h) - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6

Simplify and combine like terms.

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Step 6.1.6.1

Simplify each term.

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Step 6.1.6.1.1

Multiply $92$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{8464 + 92 (- h) - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.2

Multiply $- 1$ by $92$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - h \cdot 92 - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.3

Multiply $92$ by $- 1$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - h (- h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.4

Rewrite using the commutative property of multiplication.

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 h \cdot h) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.5

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

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Step 6.1.6.1.5.1

Move $h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 (h \cdot h)) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.5.2

Multiply $h$ by $h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h - 1 \cdot (- 1 h^{2}) - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.6

Multiply $- 1$ by $- 1$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h + 1 h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.1.7

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

$t = \frac{- 92 + h \pm \sqrt{8464 - 92 h - 92 h + h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.6.2

Subtract $92 h$ from $- 92 h$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} - 4 \cdot - 16 \cdot 24}}{2 \cdot - 16}$

Step 6.1.7

Multiply $- 4 \cdot - 16 \cdot 24$ .

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Step 6.1.7.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} + 64 \cdot 24}}{2 \cdot - 16}$

Step 6.1.7.2

Multiply $64$ by $24$ .

$t = \frac{- 92 + h \pm \sqrt{8464 - 184 h + h^{2} + 1536}}{2 \cdot - 16}$

Step 6.1.8

Add $8464$ and $1536$ .

$t = \frac{- 92 + h \pm \sqrt{- 184 h + h^{2} + 10000}}{2 \cdot - 16}$

Step 6.1.9

Reorder terms.

$t = \frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{2 \cdot - 16}$

Step 6.2

Multiply $2$ by $- 16$ .

$t = \frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{- 32}$

Step 6.3

Simplify $\frac{- 92 + h \pm \sqrt{h^{2} - 184 h + 10000}}{- 32}$ .

$t = \frac{92 - h \pm \sqrt{h^{2} - 184 h + 10000}}{32}$

Step 6.4

Change the $\pm$ to $-$ .

$t = \frac{92 - h - \sqrt{h^{2} - 184 h + 10000}}{32}$

Step 7

The final answer is the combination of both solutions.

$t = \frac{92 - h + \sqrt{h^{2} - 184 h + 10000}}{32}$

$t = \frac{92 - h - \sqrt{h^{2} - 184 h + 10000}}{32}$

Step 8

The given equation $t = \frac{92 - h + \sqrt{h^{2} - 184 h + 10000}}{32}, \frac{92 - h - \sqrt{h^{2} - 184 h + 10000}}{32}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$