Solve by Factoring 1+1/x=56/(x^2)

$1 + \frac{1}{x} = \frac{56}{x^{2}}$

Step 1

Subtract $\frac{56}{x^{2}}$ from both sides of the equation.

$1 + \frac{1}{x} - \frac{56}{x^{2}} = 0$

Step 2

Reorder terms.

$\frac{1}{x} - \frac{56}{x^{2}} + 1 = 0$

Step 3

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

$\frac{1}{x} \cdot \frac{x}{x} - \frac{56}{x^{2}} + 1 = 0$

Step 4

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x}$ by $\frac{x}{x}$ .

$\frac{x}{x \cdot x} - \frac{56}{x^{2}} + 1 = 0$

Multiply $x$ by $x$ .

$\frac{x}{x^{2}} - \frac{56}{x^{2}} + 1 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{x - 56}{x^{2}} + 1 = 0$

Step 6

Write $1$ as a fraction with a common denominator.

$\frac{x - 56}{x^{2}} + \frac{x^{2}}{x^{2}} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{x - 56 + x^{2}}{x^{2}} = 0$

Step 8

Factor $x - 56 + x^{2}$ using the AC method.

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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Write the factored form using these integers.

$\frac{(x - 7) (x + 8)}{x^{2}} = 0$

Step 9

Set the numerator equal to zero.

$(x - 7) (x + 8) = 0$

Step 10

Solve the equation for $x$ .

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

$x - 7 = 0$

$x + 8 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x - 7) (x + 8) = 0$ true.

$x = 7, - 8$

Solve by Factoring 1+1/x=56/(x^2)

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