Solve x+16/x-4=x+5

$$\frac { x + 16 } { x - 4 } = x + 5$$

Solve for x

$$x=6$$

$$x=-6$$

Steps by Finding Square Root

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by x-4.

$$x+16=\left(x-4\right)x+\left(x-4\right)\times 5$$

Use the distributive property to multiply x-4 by x.

$$x+16=x^{2}-4x+\left(x-4\right)\times 5$$

Use the distributive property to multiply x-4 by 5.

$$x+16=x^{2}-4x+5x-20$$

Combine -4x and 5x to get x.

$$x+16=x^{2}+x-20$$

Subtract x^{2} from both sides.

$$x+16-x^{2}=x-20$$

Subtract x from both sides.

$$x+16-x^{2}-x=-20$$

Combine x and -x to get 0.

$$16-x^{2}=-20$$

Subtract 16 from both sides.

$$-x^{2}=-20-16$$

Subtract 16 from -20 to get -36.

$$-x^{2}=-36$$

Divide both sides by -1.

$$x^{2}=\frac{-36}{-1}$$

Fraction \frac{-36}{-1} can be simplified to 36 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=36$$

Take the square root of both sides of the equation.

$$x=6 x=-6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by x-4.

$$x+16=\left(x-4\right)x+\left(x-4\right)\times 5$$

Use the distributive property to multiply x-4 by x.

$$x+16=x^{2}-4x+\left(x-4\right)\times 5$$

Use the distributive property to multiply x-4 by 5.

$$x+16=x^{2}-4x+5x-20$$

Combine -4x and 5x to get x.

$$x+16=x^{2}+x-20$$

Subtract x^{2} from both sides.

$$x+16-x^{2}=x-20$$

Subtract x from both sides.

$$x+16-x^{2}-x=-20$$

Combine x and -x to get 0.

$$16-x^{2}=-20$$

Add 20 to both sides.

$$16-x^{2}+20=0$$

Add 16 and 20 to get 36.

$$36-x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-x^{2}+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 36}}{2\left(-1\right)}$$

Multiply 4 times 36.

$$x=\frac{0±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{0±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±12}{-2}$$

Now solve the equation x=\frac{0±12}{-2} when ± is plus. Divide 12 by -2.

$$x=-6$$

Now solve the equation x=\frac{0±12}{-2} when ± is minus. Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-6 x=6$$