Solve x-56+64/x=0

$$x-56+ \frac{ 64 }{ x } =0$$

Solve for x

$$x=12\sqrt{5}+28\approx 54.83281573$$

$$x=28-12\sqrt{5}\approx 1.16718427$$

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.

$$xx+x\left(-56\right)+64=0$$

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

$$x^{2}+x\left(-56\right)+64=0$$

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.

$$x^{2}-56x+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -56 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 64}}{2}$$

Square -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 64}}{2}$$

Multiply -4 times 64.

$$x=\frac{-\left(-56\right)±\sqrt{3136-256}}{2}$$

Add 3136 to -256.

$$x=\frac{-\left(-56\right)±\sqrt{2880}}{2}$$

Take the square root of 2880.

$$x=\frac{-\left(-56\right)±24\sqrt{5}}{2}$$

The opposite of -56 is 56.

$$x=\frac{56±24\sqrt{5}}{2}$$

Now solve the equation x=\frac{56±24\sqrt{5}}{2} when ± is plus. Add 56 to 24\sqrt{5}.

$$x=\frac{24\sqrt{5}+56}{2}$$

Divide 56+24\sqrt{5} by 2.

$$x=12\sqrt{5}+28$$

Now solve the equation x=\frac{56±24\sqrt{5}}{2} when ± is minus. Subtract 24\sqrt{5} from 56.

$$x=\frac{56-24\sqrt{5}}{2}$$

Divide 56-24\sqrt{5} by 2.

$$x=28-12\sqrt{5}$$

The equation is now solved.

$$x=12\sqrt{5}+28 x=28-12\sqrt{5}$$

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.

$$xx+x\left(-56\right)+64=0$$

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

$$x^{2}+x\left(-56\right)+64=0$$

Subtract 64 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+x\left(-56\right)=-64$$

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.

$$x^{2}-56x=-64$$

Divide -56, the coefficient of the x term, by 2 to get -28. Then add the square of -28 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-56x+\left(-28\right)^{2}=-64+\left(-28\right)^{2}$$

Square -28.

$$x^{2}-56x+784=-64+784$$

Add -64 to 784.

$$x^{2}-56x+784=720$$

Factor x^{2}-56x+784. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-28\right)^{2}=720$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-28\right)^{2}}=\sqrt{720}$$

Simplify.

$$x-28=12\sqrt{5} x-28=-12\sqrt{5}$$

Add 28 to both sides of the equation.

$$x=12\sqrt{5}+28 x=28-12\sqrt{5}$$