Algebra Examples

$\frac{20}{2 x - 6} = \frac{x + 8}{39}$

Step 1

Cancel the common factor of $20$ and $2 x - 6$ .

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Step 1.1

Factor $2$ out of $20$ .

$\frac{2 \cdot 10}{2 x - 6} = \frac{x + 8}{39}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $2 x$ .

$\frac{2 \cdot 10}{2 (x) - 6} = \frac{x + 8}{39}$

Step 1.2.2

Factor $2$ out of $- 6$ .

$\frac{2 \cdot 10}{2 (x) + 2 (- 3)} = \frac{x + 8}{39}$

Step 1.2.3

Factor $2$ out of $2 (x) + 2 (- 3)$ .

$\frac{2 \cdot 10}{2 (x - 3)} = \frac{x + 8}{39}$

Step 1.2.4

Cancel the common factor.

$\frac{2 \cdot 10}{2 (x - 3)} = \frac{x + 8}{39}$

Step 1.2.5

Rewrite the expression.

$\frac{10}{x - 3} = \frac{x + 8}{39}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$10 \cdot 39 = (x - 3) (x + 8)$

Step 3

Solve the equation for $x$ .

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Step 3.1

Rewrite the equation as $(x - 3) (x + 8) = 10 \cdot 39$ .

$(x - 3) (x + 8) = 10 \cdot 39$

Step 3.2

Simplify $(x - 3) (x + 8)$ .

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Step 3.2.1

Expand $(x - 3) (x + 8)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

$x (x + 8) - 3 (x + 8) = 10 \cdot 39$

Step 3.2.1.2

Apply the distributive property.

$x \cdot x + x \cdot 8 - 3 (x + 8) = 10 \cdot 39$

Step 3.2.1.3

Apply the distributive property.

$x \cdot x + x \cdot 8 - 3 x - 3 \cdot 8 = 10 \cdot 39$

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 8 - 3 x - 3 \cdot 8 = 10 \cdot 39$

Step 3.2.2.1.2

Move $8$ to the left of $x$ .

$x^{2} + 8 \cdot x - 3 x - 3 \cdot 8 = 10 \cdot 39$

Step 3.2.2.1.3

Multiply $- 3$ by $8$ .

$x^{2} + 8 x - 3 x - 24 = 10 \cdot 39$

Step 3.2.2.2

Subtract $3 x$ from $8 x$ .

$x^{2} + 5 x - 24 = 10 \cdot 39$

Step 3.3

Multiply $10$ by $39$ .

$x^{2} + 5 x - 24 = 390$

Step 3.4

Subtract $390$ from both sides of the equation.

$x^{2} + 5 x - 24 - 390 = 0$

Step 3.5

Subtract $390$ from $- 24$ .

$x^{2} + 5 x - 414 = 0$

Step 3.6

Factor $x^{2} + 5 x - 414$ using the AC method.

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Step 3.6.1

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 $- 414$ and whose sum is $5$ .

$- 18, 23$

Step 3.6.2

Write the factored form using these integers.

$(x - 18) (x + 23) = 0$

Step 3.7

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

$x - 18 = 0$

$x + 23 = 0$

Step 3.8

Set $x - 18$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Step 3.8.2

Add $18$ to both sides of the equation.

$x = 18$

Step 3.9

Set $x + 23$ equal to $0$ and solve for $x$ .

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Step 3.9.1

Set $x + 23$ equal to $0$ .

$x + 23 = 0$

Step 3.9.2

Subtract $23$ from both sides of the equation.

$x = - 23$

Step 3.10

The final solution is all the values that make $(x - 18) (x + 23) = 0$ true.

$x = 18, - 23$