Algebra Examples

$1 = \frac{x^{2} + 3 x - 16}{x^{2} - 6 x + 8}$

Step 1

Rewrite the equation as $\frac{x^{2} + 3 x - 16}{x^{2} - 6 x + 8} = 1$ .

$\frac{x^{2} + 3 x - 16}{x^{2} - 6 x + 8} = 1$

Step 2

Factor $x^{2} - 6 x + 8$ using the AC method.

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Step 2.1

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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{x^{2} + 3 x - 16}{(x - 4) (x - 2)} = 1$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x - 4) (x - 2), 1$

Step 3.2

The LCM of one and any expression is the expression.

$(x - 4) (x - 2)$

Step 4

Multiply each term in $\frac{x^{2} + 3 x - 16}{(x - 4) (x - 2)} = 1$ by $(x - 4) (x - 2)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{x^{2} + 3 x - 16}{(x - 4) (x - 2)} = 1$ by $(x - 4) (x - 2)$ .

$\frac{x^{2} + 3 x - 16}{(x - 4) (x - 2)} ((x - 4) (x - 2)) = 1 ((x - 4) (x - 2))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $(x - 4) (x - 2)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x^{2} + 3 x - 16}{(x - 4) (x - 2)} ((x - 4) (x - 2)) = 1 ((x - 4) (x - 2))$

Step 4.2.1.2

Rewrite the expression.

$x^{2} + 3 x - 16 = 1 ((x - 4) (x - 2))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $(x - 4) (x - 2)$ by $1$ .

$x^{2} + 3 x - 16 = (x - 4) (x - 2)$

Step 4.3.2

Expand $(x - 4) (x - 2)$ using the FOIL Method.

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Step 4.3.2.1

Apply the distributive property.

$x^{2} + 3 x - 16 = x (x - 2) - 4 (x - 2)$

Step 4.3.2.2

Apply the distributive property.

$x^{2} + 3 x - 16 = x \cdot x + x \cdot - 2 - 4 (x - 2)$

Step 4.3.2.3

Apply the distributive property.

$x^{2} + 3 x - 16 = x \cdot x + x \cdot - 2 - 4 x - 4 \cdot - 2$

Step 4.3.3

Simplify and combine like terms.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Multiply $x$ by $x$ .

$x^{2} + 3 x - 16 = x^{2} + x \cdot - 2 - 4 x - 4 \cdot - 2$

Step 4.3.3.1.2

Move $- 2$ to the left of $x$ .

$x^{2} + 3 x - 16 = x^{2} - 2 \cdot x - 4 x - 4 \cdot - 2$

Step 4.3.3.1.3

Multiply $- 4$ by $- 2$ .

$x^{2} + 3 x - 16 = x^{2} - 2 x - 4 x + 8$

Step 4.3.3.2

Subtract $4 x$ from $- 2 x$ .

$x^{2} + 3 x - 16 = x^{2} - 6 x + 8$

Step 5

Solve the equation.

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Step 5.1

Move all terms containing $x$ to the left side of the equation.

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Step 5.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 3 x - 16 - x^{2} = - 6 x + 8$

Step 5.1.2

Add $6 x$ to both sides of the equation.

$x^{2} + 3 x - 16 - x^{2} + 6 x = 8$

Step 5.1.3

Combine the opposite terms in $x^{2} + 3 x - 16 - x^{2} + 6 x$ .

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Step 5.1.3.1

Subtract $x^{2}$ from $x^{2}$ .

$3 x - 16 + 0 + 6 x = 8$

Step 5.1.3.2

Add $3 x - 16$ and $0$ .

$3 x - 16 + 6 x = 8$

Step 5.1.4

Add $3 x$ and $6 x$ .

$9 x - 16 = 8$

Step 5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 5.2.1

Add $16$ to both sides of the equation.

$9 x = 8 + 16$

Step 5.2.2

Add $8$ and $16$ .

$9 x = 24$

Step 5.3

Divide each term in $9 x = 24$ by $9$ and simplify.

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Step 5.3.1

Divide each term in $9 x = 24$ by $9$ .

$\frac{9 x}{9} = \frac{24}{9}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $9$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{24}{9}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{24}{9}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $24$ and $9$ .

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Step 5.3.3.1.1

Factor $3$ out of $24$ .

$x = \frac{3 (8)}{9}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 8}{3 \cdot 3}$

Step 5.3.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 8}{3 \cdot 3}$

Step 5.3.3.1.2.3

Rewrite the expression.

$x = \frac{8}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}$

Decimal Form:

$x = 2.\overline{6}$

Mixed Number Form:

$x = 2 \frac{2}{3}$