Solve (1230-x)(1231-x)=500

$$( 1230 - x ) ( 1231 - x ) = 500$$

Solve for x

$$x = \frac{2461 - \sqrt{2001}}{2} \approx 1208.133730754$$

$$x = \frac{\sqrt{2001} + 2461}{2} \approx 1252.866269246$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1230-x by 1231-x and combine like terms.

$$1514130-2461x+x^{2}=500$$

Subtract 500 from both sides.

$$1514130-2461x+x^{2}-500=0$$

Subtract 500 from 1514130 to get 1513630.

$$1513630-2461x+x^{2}=0$$

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.

$$x^{2}-2461x+1513630=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2461 for b, and 1513630 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2461\right)±\sqrt{\left(-2461\right)^{2}-4\times 1513630}}{2}$$

Square -2461.

$$x=\frac{-\left(-2461\right)±\sqrt{6056521-4\times 1513630}}{2}$$

Multiply -4 times 1513630.

$$x=\frac{-\left(-2461\right)±\sqrt{6056521-6054520}}{2}$$

Add 6056521 to -6054520.

$$x=\frac{-\left(-2461\right)±\sqrt{2001}}{2}$$

The opposite of -2461 is 2461.

$$x=\frac{2461±\sqrt{2001}}{2}$$

Now solve the equation x=\frac{2461±\sqrt{2001}}{2} when ± is plus. Add 2461 to \sqrt{2001}.

$$x=\frac{\sqrt{2001}+2461}{2}$$

Now solve the equation x=\frac{2461±\sqrt{2001}}{2} when ± is minus. Subtract \sqrt{2001} from 2461.

$$x=\frac{2461-\sqrt{2001}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2001}+2461}{2} x=\frac{2461-\sqrt{2001}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 1230-x by 1231-x and combine like terms.

$$1514130-2461x+x^{2}=500$$

Subtract 1514130 from both sides.

$$-2461x+x^{2}=500-1514130$$

Subtract 1514130 from 500 to get -1513630.

$$-2461x+x^{2}=-1513630$$

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.

$$x^{2}-2461x=-1513630$$

Divide -2461, the coefficient of the x term, by 2 to get -\frac{2461}{2}. Then add the square of -\frac{2461}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2461x+\left(-\frac{2461}{2}\right)^{2}=-1513630+\left(-\frac{2461}{2}\right)^{2}$$

Square -\frac{2461}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-2461x+\frac{6056521}{4}=-1513630+\frac{6056521}{4}$$

Add -1513630 to \frac{6056521}{4}.

$$x^{2}-2461x+\frac{6056521}{4}=\frac{2001}{4}$$

Factor x^{2}-2461x+\frac{6056521}{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}.

$$\left(x-\frac{2461}{2}\right)^{2}=\frac{2001}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2461}{2}\right)^{2}}=\sqrt{\frac{2001}{4}}$$

Simplify.

$$x-\frac{2461}{2}=\frac{\sqrt{2001}}{2} x-\frac{2461}{2}=-\frac{\sqrt{2001}}{2}$$

Add \frac{2461}{2} to both sides of the equation.

$$x=\frac{\sqrt{2001}+2461}{2} x=\frac{2461-\sqrt{2001}}{2}$$