Finite Math Examples

$\frac{1}{2} \cdot ((h + 3) (h)) = 27$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot ((h + 3) h)) = 2 \cdot 27$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot ((h + 3) h))$ .

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Step 2.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (h \cdot h + 3 h)) = 2 \cdot 27$

Step 2.1.1.2

Multiply $h$ by $h$ .

$2 (\frac{1}{2} \cdot (h^{2} + 3 h)) = 2 \cdot 27$

Step 2.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} h^{2} + \frac{1}{2} (3 h)) = 2 \cdot 27$

Step 2.1.1.4

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

$2 (\frac{h^{2}}{2} + \frac{1}{2} (3 h)) = 2 \cdot 27$

Step 2.1.1.5

Multiply $\frac{1}{2} (3 h)$ .

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Step 2.1.1.5.1

Combine $3$ and $\frac{1}{2}$ .

$2 (\frac{h^{2}}{2} + \frac{3}{2} h) = 2 \cdot 27$

Step 2.1.1.5.2

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

$2 (\frac{h^{2}}{2} + \frac{3 h}{2}) = 2 \cdot 27$

Step 2.1.1.6

Apply the distributive property.

$2 \frac{h^{2}}{2} + 2 \frac{3 h}{2} = 2 \cdot 27$

Step 2.1.1.7

Cancel the common factor of $2$ .

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Step 2.1.1.7.1

Cancel the common factor.

$2 \frac{h^{2}}{2} + 2 \frac{3 h}{2} = 2 \cdot 27$

Step 2.1.1.7.2

Rewrite the expression.

$h^{2} + 2 \frac{3 h}{2} = 2 \cdot 27$

Step 2.1.1.8

Cancel the common factor of $2$ .

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Step 2.1.1.8.1

Cancel the common factor.

$h^{2} + 2 \frac{3 h}{2} = 2 \cdot 27$

Step 2.1.1.8.2

Rewrite the expression.

$h^{2} + 3 h = 2 \cdot 27$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2$ by $27$ .

$h^{2} + 3 h = 54$

Step 3

Subtract $54$ from both sides of the equation.

$h^{2} + 3 h - 54 = 0$

Step 4

Factor $h^{2} + 3 h - 54$ using the AC method.

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Step 4.1

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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Step 4.2

Write the factored form using these integers.

$(h - 6) (h + 9) = 0$

Step 5

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

$h - 6 = 0$

$h + 9 = 0$

Step 6

Set $h - 6$ equal to $0$ and solve for $h$ .

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Step 6.1

Set $h - 6$ equal to $0$ .

$h - 6 = 0$

Step 6.2

Add $6$ to both sides of the equation.

$h = 6$

Step 7

Set $h + 9$ equal to $0$ and solve for $h$ .

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Step 7.1

Set $h + 9$ equal to $0$ .

$h + 9 = 0$

Step 7.2

Subtract $9$ from both sides of the equation.

$h = - 9$

Step 8

The final solution is all the values that make $(h - 6) (h + 9) = 0$ true.

$h = 6, - 9$