Solve x/x+x/8=x/16

$$\frac { x } { x } + \frac { x } { 8 } = \frac { x } { 16 }$$

Solve for x

$$x=-16$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16x, the least common multiple of x,8,16.

$$16x+2xx=xx$$

Multiply x and x to get x^{2}.

$$16x+2xx=x^{2}$$

Multiply x and x to get x^{2}.

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

Subtract x^{2} from both sides.

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

Combine 2x^{2} and -x^{2} to get x^{2}.

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

Factor out x.

$$x\left(16+x\right)=0$$

To find equation solutions, solve x=0 and 16+x=0.

$$x=0 x=-16$$

Variable x cannot be equal to 0.

$$x=-16$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16x, the least common multiple of x,8,16.

$$16x+2xx=xx$$

Multiply x and x to get x^{2}.

$$16x+2xx=x^{2}$$

Multiply x and x to get x^{2}.

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

Subtract x^{2} from both sides.

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

Combine 2x^{2} and -x^{2} to get x^{2}.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}}}{2}$$

Take the square root of 16^{2}.

$$x=\frac{-16±16}{2}$$

Now solve the equation x=\frac{-16±16}{2} when ± is plus. Add -16 to 16.

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

Now solve the equation x=\frac{-16±16}{2} when ± is minus. Subtract 16 from -16.

$$x=-\frac{32}{2}$$

Divide -32 by 2.

$$x=-16$$

The equation is now solved.

$$x=0 x=-16$$

Variable x cannot be equal to 0.

$$x=-16$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16x, the least common multiple of x,8,16.

$$16x+2xx=xx$$

Multiply x and x to get x^{2}.

$$16x+2xx=x^{2}$$

Multiply x and x to get x^{2}.

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

Subtract x^{2} from both sides.

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

Combine 2x^{2} and -x^{2} to get x^{2}.

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

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.

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

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=8^{2}$$

Square 8.

$$x^{2}+16x+64=64$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{64}$$

Simplify.

$$x+8=8 x+8=-8$$

Subtract 8 from both sides of the equation.

$$x=0 x=-16$$

Variable x cannot be equal to 0.

$$x=-16$$