Finite Math Examples

$35 = \frac{1}{2} \cdot ((x + 3) x)$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot ((x + 3) x) = 35$ .

$\frac{1}{2} \cdot ((x + 3) x) = 35$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot ((x + 3) x)) = 2 \cdot 35$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x \cdot x + 3 x)) = 2 \cdot 35$

Step 3.1.1.2

Multiply $x$ by $x$ .

$2 (\frac{1}{2} \cdot (x^{2} + 3 x)) = 2 \cdot 35$

Step 3.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} x^{2} + \frac{1}{2} (3 x)) = 2 \cdot 35$

Step 3.1.1.4

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

$2 (\frac{x^{2}}{2} + \frac{1}{2} (3 x)) = 2 \cdot 35$

Step 3.1.1.5

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

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Step 3.1.1.5.1

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

$2 (\frac{x^{2}}{2} + \frac{3}{2} x) = 2 \cdot 35$

Step 3.1.1.5.2

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

$2 (\frac{x^{2}}{2} + \frac{3 x}{2}) = 2 \cdot 35$

Step 3.1.1.6

Apply the distributive property.

$2 \frac{x^{2}}{2} + 2 \frac{3 x}{2} = 2 \cdot 35$

Step 3.1.1.7

Cancel the common factor of $2$ .

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Step 3.1.1.7.1

Cancel the common factor.

$2 \frac{x^{2}}{2} + 2 \frac{3 x}{2} = 2 \cdot 35$

Step 3.1.1.7.2

Rewrite the expression.

$x^{2} + 2 \frac{3 x}{2} = 2 \cdot 35$

Step 3.1.1.8

Cancel the common factor of $2$ .

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Step 3.1.1.8.1

Cancel the common factor.

$x^{2} + 2 \frac{3 x}{2} = 2 \cdot 35$

Step 3.1.1.8.2

Rewrite the expression.

$x^{2} + 3 x = 2 \cdot 35$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $35$ .

$x^{2} + 3 x = 70$

Step 4

Subtract $70$ from both sides of the equation.

$x^{2} + 3 x - 70 = 0$

Step 5

Factor $x^{2} + 3 x - 70$ using the AC method.

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Step 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 $- 70$ and whose sum is $3$ .

$- 7, 10$

Step 5.2

Write the factored form using these integers.

$(x - 7) (x + 10) = 0$

Step 6

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

$x - 7 = 0$

$x + 10 = 0$

Step 7

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 7.2

Add $7$ to both sides of the equation.

$x = 7$

Step 8

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 8.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 9

The final solution is all the values that make $(x - 7) (x + 10) = 0$ true.

$x = 7, - 10$