Solve 1/x+10=x+4/x

$$\frac { 1 } { x + 10 } = \frac { x + 4 } { x }$$

Solve for x

$$x=-8$$

$$x=-5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x+10,x.

$$x=\left(x+10\right)\left(x+4\right)$$

Use the distributive property to multiply x+10 by x+4 and combine like terms.

$$x=x^{2}+14x+40$$

Subtract x^{2} from both sides.

$$x-x^{2}=14x+40$$

Subtract 14x from both sides.

$$x-x^{2}-14x=40$$

Combine x and -14x to get -13x.

$$-13x-x^{2}=40$$

Subtract 40 from both sides.

$$-13x-x^{2}-40=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-13x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -13 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-1\right)\left(-40\right)}}{2\left(-1\right)}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-1\right)\left(-40\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-13\right)±\sqrt{169+4\left(-40\right)}}{2\left(-1\right)}$$

Multiply 4 times -40.

$$x=\frac{-\left(-13\right)±\sqrt{169-160}}{2\left(-1\right)}$$

Add 169 to -160.

$$x=\frac{-\left(-13\right)±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

$$x=\frac{-\left(-13\right)±3}{2\left(-1\right)}$$

The opposite of -13 is 13.

$$x=\frac{13±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{13±3}{-2}$$

Now solve the equation x=\frac{13±3}{-2} when ± is plus. Add 13 to 3.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{13±3}{-2} when ± is minus. Subtract 3 from 13.

$$x=\frac{10}{-2}$$

Divide 10 by -2.

$$x=-5$$

The equation is now solved.

$$x=-8 x=-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x+10,x.

$$x=\left(x+10\right)\left(x+4\right)$$

Use the distributive property to multiply x+10 by x+4 and combine like terms.

$$x=x^{2}+14x+40$$

Subtract x^{2} from both sides.

$$x-x^{2}=14x+40$$

Subtract 14x from both sides.

$$x-x^{2}-14x=40$$

Combine x and -14x to get -13x.

$$-13x-x^{2}=40$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}-13x=40$$

Divide both sides by -1.

$$\frac{-x^{2}-13x}{-1}=\frac{40}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{13}{-1}\right)x=\frac{40}{-1}$$

Divide -13 by -1.

$$x^{2}+13x=\frac{40}{-1}$$

Divide 40 by -1.

$$x^{2}+13x=-40$$

Divide 13, the coefficient of the x term, by 2 to get \frac{13}{2}. Then add the square of \frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=-40+\left(\frac{13}{2}\right)^{2}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+13x+\frac{169}{4}=-40+\frac{169}{4}$$

Add -40 to \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=\frac{9}{4}$$

Factor x^{2}+13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{13}{2}=\frac{3}{2} x+\frac{13}{2}=-\frac{3}{2}$$

Subtract \frac{13}{2} from both sides of the equation.

$$x=-5 x=-8$$