Algebra Examples

$\frac{x^{2} + x - 2}{x - 3} = - 3$

Step 1

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

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Step 1.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$x - 3, 1$

Step 2.2

Remove parentheses.

$x - 3, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x - 3$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x - 3$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

$(x - 1) (x + 2) = - 3 (x - 3)$

Step 3.2.2

Expand $(x - 1) (x + 2)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$x (x + 2) - 1 (x + 2) = - 3 (x - 3)$

Step 3.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 2 - 1 (x + 2) = - 3 (x - 3)$

Step 3.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 2 - 1 x - 1 \cdot 2 = - 3 (x - 3)$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $x$ by $x$ .

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

Step 3.2.3.1.2

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

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

Step 3.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.2.3.1.4

Multiply $- 1$ by $2$ .

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

Step 3.2.3.2

Subtract $x$ from $2 x$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Multiply $- 3$ by $- 3$ .

$x^{2} + x - 2 = - 3 x + 9$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Add $3 x$ to both sides of the equation.

$x^{2} + x - 2 + 3 x = 9$

Step 4.1.2

Add $x$ and $3 x$ .

$x^{2} + 4 x - 2 = 9$

Step 4.2

Move all terms to the left side of the equation and simplify.

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Step 4.2.1

Subtract $9$ from both sides of the equation.

$x^{2} + 4 x - 2 - 9 = 0$

Step 4.2.2

Subtract $9$ from $- 2$ .

$x^{2} + 4 x - 11 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 1$ , $b = 4$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 11$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 11}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $- 11$ .

$x = \frac{- 4 \pm \sqrt{16 + 44}}{2 \cdot 1}$

Step 4.5.1.3

Add $16$ and $44$ .

$x = \frac{- 4 \pm \sqrt{60}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 4.5.1.4.1

Factor $4$ out of $60$ .

$x = \frac{- 4 \pm \sqrt{4 (15)}}{2 \cdot 1}$

Step 4.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{15}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{15}}{2}$

Step 4.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{15}}{2}$ .

$x = - 2 \pm \sqrt{15}$

Step 4.6

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{15}, - 2 - \sqrt{15}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \sqrt{15}, - 2 - \sqrt{15}$

Decimal Form:

$x = 1.87298334 \dots, - 5.87298334 \dots$