Algebra Examples

$x + \frac{6}{x} = - 5$

Step 1

Add $5$ to both sides of the equation.

$x + \frac{6}{x} + 5 = 0$

Step 2

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x \cdot x}{x} + \frac{6}{x} + 5 = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{x \cdot x + 6}{x} + 5 = 0$

Step 4

Multiply $x$ by $x$ .

$\frac{x^{2} + 6}{x} + 5 = 0$

Step 5

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

$\frac{x^{2} + 6}{x} + \frac{5 x}{x} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{x^{2} + 6 + 5 x}{x} = 0$

Step 7

Factor $x^{2} + 6 + 5 x$ using the AC method.

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Step 7.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 7.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 3)}{x} = 0$

Step 8

Set the numerator equal to zero.

$(x + 2) (x + 3) = 0$

Step 9

Solve the equation for $x$ .

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Step 9.1

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

$x + 2 = 0$

$x + 3 = 0$

Step 9.2

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 9.2.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 9.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9.3

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 9.3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 9.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 9.4

The final solution is all the values that make $(x + 2) (x + 3) = 0$ true.

$x = - 2, - 3$