Solve x+84=160divx

$$x+84=160 \div x$$

Solve for x

$$x=2\sqrt{481}-42\approx 1.863424399$$

$$x=-2\sqrt{481}-42\approx -85.863424399$$

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 x.

$$xx+x\times 84=160$$

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

$$x^{2}+x\times 84=160$$

Subtract 160 from both sides.

$$x^{2}+x\times 84-160=0$$

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.

$$x^{2}+84x-160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 84 for b, and -160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-84±\sqrt{84^{2}-4\left(-160\right)}}{2}$$

Square 84.

$$x=\frac{-84±\sqrt{7056-4\left(-160\right)}}{2}$$

Multiply -4 times -160.

$$x=\frac{-84±\sqrt{7056+640}}{2}$$

Add 7056 to 640.

$$x=\frac{-84±\sqrt{7696}}{2}$$

Take the square root of 7696.

$$x=\frac{-84±4\sqrt{481}}{2}$$

Now solve the equation x=\frac{-84±4\sqrt{481}}{2} when ± is plus. Add -84 to 4\sqrt{481}.

$$x=\frac{4\sqrt{481}-84}{2}$$

Divide -84+4\sqrt{481} by 2.

$$x=2\sqrt{481}-42$$

Now solve the equation x=\frac{-84±4\sqrt{481}}{2} when ± is minus. Subtract 4\sqrt{481} from -84.

$$x=\frac{-4\sqrt{481}-84}{2}$$

Divide -84-4\sqrt{481} by 2.

$$x=-2\sqrt{481}-42$$

The equation is now solved.

$$x=2\sqrt{481}-42 x=-2\sqrt{481}-42$$

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 x.

$$xx+x\times 84=160$$

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

$$x^{2}+x\times 84=160$$

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.

$$x^{2}+84x=160$$

Divide 84, the coefficient of the x term, by 2 to get 42. Then add the square of 42 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+84x+42^{2}=160+42^{2}$$

Square 42.

$$x^{2}+84x+1764=160+1764$$

Add 160 to 1764.

$$x^{2}+84x+1764=1924$$

Factor x^{2}+84x+1764. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+42\right)^{2}=1924$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+42\right)^{2}}=\sqrt{1924}$$

Simplify.

$$x+42=2\sqrt{481} x+42=-2\sqrt{481}$$

Subtract 42 from both sides of the equation.

$$x=2\sqrt{481}-42 x=-2\sqrt{481}-42$$