Finite Math Examples

$\frac{1}{2} \cdot ((h - 6) (h)) = 416$

Step 1

Subtract $416$ from both sides of the equation.

$\frac{1}{2} \cdot ((h - 6) (h)) - 416 = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\frac{1}{2} \cdot (h \cdot h - 6 h) - 416 = 0$

Step 2.2

Multiply $h$ by $h$ .

$\frac{1}{2} \cdot (h^{2} - 6 h) - 416 = 0$

Step 2.3

Apply the distributive property.

$\frac{1}{2} h^{2} + \frac{1}{2} (- 6 h) - 416 = 0$

Step 2.4

Combine $\frac{1}{2}$ and $h^{2}$ .

$\frac{h^{2}}{2} + \frac{1}{2} (- 6 h) - 416 = 0$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Factor $2$ out of $- 6 h$ .

$\frac{h^{2}}{2} + \frac{1}{2} (2 (- 3 h)) - 416 = 0$

Step 2.5.2

Cancel the common factor.

$\frac{h^{2}}{2} + \frac{1}{2} (2 (- 3 h)) - 416 = 0$

Step 2.5.3

Rewrite the expression.

$\frac{h^{2}}{2} - 3 h - 416 = 0$

Step 3

To write $- 3 h$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{h^{2}}{2} - 3 h \cdot \frac{2}{2} - 416 = 0$

Step 4

Combine $- 3 h$ and $\frac{2}{2}$ .

$\frac{h^{2}}{2} + \frac{- 3 h \cdot 2}{2} - 416 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{h^{2} - 3 h \cdot 2}{2} - 416 = 0$

Step 6

Simplify the numerator.

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Step 6.1

Factor $h$ out of $h^{2} - 3 h \cdot 2$ .

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Step 6.1.1

Factor $h$ out of $h^{2}$ .

$\frac{h \cdot h - 3 h \cdot 2}{2} - 416 = 0$

Step 6.1.2

Factor $h$ out of $- 3 h \cdot 2$ .

$\frac{h \cdot h + h (- 3 \cdot 2)}{2} - 416 = 0$

Step 6.1.3

Factor $h$ out of $h \cdot h + h (- 3 \cdot 2)$ .

$\frac{h (h - 3 \cdot 2)}{2} - 416 = 0$

Step 6.2

Multiply $- 3$ by $2$ .

$\frac{h (h - 6)}{2} - 416 = 0$

Step 7

To write $- 416$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{h (h - 6)}{2} - 416 \cdot \frac{2}{2} = 0$

Step 8

Combine $- 416$ and $\frac{2}{2}$ .

$\frac{h (h - 6)}{2} + \frac{- 416 \cdot 2}{2} = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{h (h - 6) - 416 \cdot 2}{2} = 0$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{h \cdot h + h \cdot - 6 - 416 \cdot 2}{2} = 0$

Step 10.2

Multiply $h$ by $h$ .

$\frac{h^{2} + h \cdot - 6 - 416 \cdot 2}{2} = 0$

Step 10.3

Move $- 6$ to the left of $h$ .

$\frac{h^{2} - 6 h - 416 \cdot 2}{2} = 0$

Step 10.4

Multiply $- 416$ by $2$ .

$\frac{h^{2} - 6 h - 832}{2} = 0$

Step 10.5

Factor $h^{2} - 6 h - 832$ using the AC method.

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Step 10.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 832$ and whose sum is $- 6$ .

$- 32, 26$

Step 10.5.2

Write the factored form using these integers.

$\frac{(h - 32) (h + 26)}{2} = 0$

Step 11

Set the numerator equal to zero.

$(h - 32) (h + 26) = 0$

Step 12

Solve the equation for $h$ .

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Step 12.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$h - 32 = 0$

$h + 26 = 0$

Step 12.2

Set $h - 32$ equal to $0$ and solve for $h$ .

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Step 12.2.1

Set $h - 32$ equal to $0$ .

$h - 32 = 0$

Step 12.2.2

Add $32$ to both sides of the equation.

$h = 32$

Step 12.3

Set $h + 26$ equal to $0$ and solve for $h$ .

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Step 12.3.1

Set $h + 26$ equal to $0$ .

$h + 26 = 0$

Step 12.3.2

Subtract $26$ from both sides of the equation.

$h = - 26$

Step 12.4

The final solution is all the values that make $(h - 32) (h + 26) = 0$ true.

$h = 32, - 26$