Solve by Factoring 10/x=x-3

$\frac{10}{x} = x - 3$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$\frac{10}{x} - x = - 3$

Add $3$ to both sides of the equation.

$\frac{10}{x} - x + 3 = 0$

Step 2

Reorder terms.

$- x + \frac{10}{x} + 3 = 0$

Step 3

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

$- x \cdot \frac{x}{x} + \frac{10}{x} + 3 = 0$

Step 4

Combine $- x$ and $\frac{x}{x}$ .

$\frac{- x \cdot x}{x} + \frac{10}{x} + 3 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- x \cdot x + 10}{x} + 3 = 0$

Step 6

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

Move $x$ .

$\frac{- (x \cdot x) + 10}{x} + 3 = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} + 10}{x} + 3 = 0$

Step 7

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

$\frac{- x^{2} + 10}{x} + \frac{3 x}{x} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{- x^{2} + 10 + 3 x}{x} = 0$

Step 9

Factor by grouping.

Reorder terms.

$\frac{- x^{2} + 3 x + 10}{x} = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 10 = - 10$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

$\frac{- x^{2} + 3 (x) + 10}{x} = 0$

Rewrite $3$ as $- 2$ plus $5$

$\frac{- x^{2} + (- 2 + 5) x + 10}{x} = 0$

Apply the distributive property.

$\frac{- x^{2} - 2 x + 5 x + 10}{x} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 2 x) + 5 x + 10}{x} = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x - 2) - 5 (- x - 2)}{x} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$\frac{(- x - 2) (x - 5)}{x} = 0$

Step 10

Set the numerator equal to zero.

$(- x - 2) (x - 5) = 0$

Step 11

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 $0$ .

$- x - 2 = 0$

$x - 5 = 0$

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

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

$- x - 2 = 0$

Solve $- x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$- x = 2$

Divide each term in $- x = 2$ by $- 1$ and simplify.

Divide each term in $- x = 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$x = - 2$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(- x - 2) (x - 5) = 0$ true.

$x = - 2, 5$

Solve by Factoring 10/x=x-3

Related Questions

Support LCMGCF.com by disabling your adblocker.