Solve 1200/x=x+20

$$\frac{ 1200 }{ x } =x+20$$

Solve for x

$$x=10\sqrt{13}-10\approx 26.055512755$$

$$x=-10\sqrt{13}-10\approx -46.055512755$$

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.

$$1200=xx+x\times 20$$

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

$$1200=x^{2}+x\times 20$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+x\times 20=1200$$

Subtract 1200 from both sides.

$$x^{2}+x\times 20-1200=0$$

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.

$$x^{2}+20x-1200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -1200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-1200\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-1200\right)}}{2}$$

Multiply -4 times -1200.

$$x=\frac{-20±\sqrt{400+4800}}{2}$$

Add 400 to 4800.

$$x=\frac{-20±\sqrt{5200}}{2}$$

Take the square root of 5200.

$$x=\frac{-20±20\sqrt{13}}{2}$$

Now solve the equation x=\frac{-20±20\sqrt{13}}{2} when ± is plus. Add -20 to 20\sqrt{13}.

$$x=\frac{20\sqrt{13}-20}{2}$$

Divide -20+20\sqrt{13} by 2.

$$x=10\sqrt{13}-10$$

Now solve the equation x=\frac{-20±20\sqrt{13}}{2} when ± is minus. Subtract 20\sqrt{13} from -20.

$$x=\frac{-20\sqrt{13}-20}{2}$$

Divide -20-20\sqrt{13} by 2.

$$x=-10\sqrt{13}-10$$

The equation is now solved.

$$x=10\sqrt{13}-10 x=-10\sqrt{13}-10$$

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.

$$1200=xx+x\times 20$$

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

$$1200=x^{2}+x\times 20$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+x\times 20=1200$$

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.

$$x^{2}+20x=1200$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=1200+10^{2}$$

Square 10.

$$x^{2}+20x+100=1200+100$$

Add 1200 to 100.

$$x^{2}+20x+100=1300$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=1300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{1300}$$

Simplify.

$$x+10=10\sqrt{13} x+10=-10\sqrt{13}$$

Subtract 10 from both sides of the equation.

$$x=10\sqrt{13}-10 x=-10\sqrt{13}-10$$