Algebra Examples

$\frac{- 3}{x + 2} + \frac{5 x}{x - 1} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1

Simplify $\frac{- 3}{x + 2} + \frac{5 x}{x - 1}$ .

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Step 1.1

Move the negative in front of the fraction.

$- \frac{3}{x + 2} + \frac{5 x}{x - 1} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.2

To write $- \frac{3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$- \frac{3}{x + 2} \cdot \frac{x - 1}{x - 1} + \frac{5 x}{x - 1} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.3

To write $\frac{5 x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$- \frac{3}{x + 2} \cdot \frac{x - 1}{x - 1} + \frac{5 x}{x - 1} \cdot \frac{x + 2}{x + 2} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.4

Write each expression with a common denominator of $(x + 2) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{3}{x + 2}$ by $\frac{x - 1}{x - 1}$ .

$- \frac{3 (x - 1)}{(x + 2) (x - 1)} + \frac{5 x}{x - 1} \cdot \frac{x + 2}{x + 2} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.4.2

Multiply $\frac{5 x}{x - 1}$ by $\frac{x + 2}{x + 2}$ .

$- \frac{3 (x - 1)}{(x + 2) (x - 1)} + \frac{5 x (x + 2)}{(x - 1) (x + 2)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.4.3

Reorder the factors of $(x - 1) (x + 2)$ .

$- \frac{3 (x - 1)}{(x + 2) (x - 1)} + \frac{5 x (x + 2)}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{- 3 (x - 1) + 5 x (x + 2)}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

$\frac{- 3 x - 3 \cdot - 1 + 5 x (x + 2)}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.2

Multiply $- 3$ by $- 1$ .

$\frac{- 3 x + 3 + 5 x (x + 2)}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.3

Apply the distributive property.

$\frac{- 3 x + 3 + 5 x \cdot x + 5 x \cdot 2}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.6.4.1

Move $x$ .

$\frac{- 3 x + 3 + 5 (x \cdot x) + 5 x \cdot 2}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.4.2

Multiply $x$ by $x$ .

$\frac{- 3 x + 3 + 5 x^{2} + 5 x \cdot 2}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.5

Multiply $2$ by $5$ .

$\frac{- 3 x + 3 + 5 x^{2} + 10 x}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.6

Add $- 3 x$ and $10 x$ .

$\frac{7 x + 3 + 5 x^{2}}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 1.6.7

Reorder terms.

$\frac{5 x^{2} + 7 x + 3}{(x + 2) (x - 1)} = \frac{x + 2}{x^{2} - 3 x + 2}$

Step 2

Factor $x^{2} - 3 x + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 2.2

Write the factored form using these integers.

$\frac{5 x^{2} + 7 x + 3}{(x + 2) (x - 1)} = \frac{x + 2}{(x - 2) (x - 1)}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 2.39846599$

Step 4