Solve 0=11250-525x+x^2

$$0 = 11250 - 525 x + x ^ { 2 }$$

Solve for x

$$x = \frac{75 \sqrt{41} + 525}{2} \approx 502.617158904$$

$$x = \frac{525 - 75 \sqrt{41}}{2} \approx 22.382841096$$

Steps Using the Quadratic Formula

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

$$11250-525x+x^{2}=0$$

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.

$$x^{2}-525x+11250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -525 for b, and 11250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-525\right)±\sqrt{\left(-525\right)^{2}-4\times 11250}}{2}$$

Square -525.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4\times 11250}}{2}$$

Multiply -4 times 11250.

$$x=\frac{-\left(-525\right)±\sqrt{275625-45000}}{2}$$

Add 275625 to -45000.

$$x=\frac{-\left(-525\right)±\sqrt{230625}}{2}$$

Take the square root of 230625.

$$x=\frac{-\left(-525\right)±75\sqrt{41}}{2}$$

The opposite of -525 is 525.

$$x=\frac{525±75\sqrt{41}}{2}$$

Now solve the equation x=\frac{525±75\sqrt{41}}{2} when ± is plus. Add 525 to 75\sqrt{41}.

$$x=\frac{75\sqrt{41}+525}{2}$$

Now solve the equation x=\frac{525±75\sqrt{41}}{2} when ± is minus. Subtract 75\sqrt{41} from 525.

$$x=\frac{525-75\sqrt{41}}{2}$$

The equation is now solved.

$$x=\frac{75\sqrt{41}+525}{2} x=\frac{525-75\sqrt{41}}{2}$$

Steps for Completing the Square

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

$$11250-525x+x^{2}=0$$

Subtract 11250 from both sides. Anything subtracted from zero gives its negation.

$$-525x+x^{2}=-11250$$

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.

$$x^{2}-525x=-11250$$

Divide -525, the coefficient of the x term, by 2 to get -\frac{525}{2}. Then add the square of -\frac{525}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-525x+\left(-\frac{525}{2}\right)^{2}=-11250+\left(-\frac{525}{2}\right)^{2}$$

Square -\frac{525}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-525x+\frac{275625}{4}=-11250+\frac{275625}{4}$$

Add -11250 to \frac{275625}{4}.

$$x^{2}-525x+\frac{275625}{4}=\frac{230625}{4}$$

Factor x^{2}-525x+\frac{275625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{525}{2}\right)^{2}=\frac{230625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{525}{2}\right)^{2}}=\sqrt{\frac{230625}{4}}$$

Simplify.

$$x-\frac{525}{2}=\frac{75\sqrt{41}}{2} x-\frac{525}{2}=-\frac{75\sqrt{41}}{2}$$

Add \frac{525}{2} to both sides of the equation.

$$x=\frac{75\sqrt{41}+525}{2} x=\frac{525-75\sqrt{41}}{2}$$