Solve (2008-a)(2006-a)=2007

$$( 2008 - a ) ( 2006 - a ) = 2007$$

Solve for a

$$a=2007-2\sqrt{502}\approx 1962.189286995$$

$$a=2\sqrt{502}+2007\approx 2051.810713005$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2008-a by 2006-a and combine like terms.

$$4028048-4014a+a^{2}=2007$$

Subtract 2007 from both sides.

$$4028048-4014a+a^{2}-2007=0$$

Subtract 2007 from 4028048 to get 4026041.

$$4026041-4014a+a^{2}=0$$

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.

$$a^{2}-4014a+4026041=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4014 for b, and 4026041 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4014\right)±\sqrt{\left(-4014\right)^{2}-4\times 4026041}}{2}$$

Square -4014.

$$a=\frac{-\left(-4014\right)±\sqrt{16112196-4\times 4026041}}{2}$$

Multiply -4 times 4026041.

$$a=\frac{-\left(-4014\right)±\sqrt{16112196-16104164}}{2}$$

Add 16112196 to -16104164.

$$a=\frac{-\left(-4014\right)±\sqrt{8032}}{2}$$

Take the square root of 8032.

$$a=\frac{-\left(-4014\right)±4\sqrt{502}}{2}$$

The opposite of -4014 is 4014.

$$a=\frac{4014±4\sqrt{502}}{2}$$

Now solve the equation a=\frac{4014±4\sqrt{502}}{2} when ± is plus. Add 4014 to 4\sqrt{502}.

$$a=\frac{4\sqrt{502}+4014}{2}$$

Divide 4014+4\sqrt{502} by 2.

$$a=2\sqrt{502}+2007$$

Now solve the equation a=\frac{4014±4\sqrt{502}}{2} when ± is minus. Subtract 4\sqrt{502} from 4014.

$$a=\frac{4014-4\sqrt{502}}{2}$$

Divide 4014-4\sqrt{502} by 2.

$$a=2007-2\sqrt{502}$$

The equation is now solved.

$$a=2\sqrt{502}+2007 a=2007-2\sqrt{502}$$

Steps for Completing the Square

Use the distributive property to multiply 2008-a by 2006-a and combine like terms.

$$4028048-4014a+a^{2}=2007$$

Subtract 4028048 from both sides.

$$-4014a+a^{2}=2007-4028048$$

Subtract 4028048 from 2007 to get -4026041.

$$-4014a+a^{2}=-4026041$$

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.

$$a^{2}-4014a=-4026041$$

Divide -4014, the coefficient of the x term, by 2 to get -2007. Then add the square of -2007 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-4014a+\left(-2007\right)^{2}=-4026041+\left(-2007\right)^{2}$$

Square -2007.

$$a^{2}-4014a+4028049=-4026041+4028049$$

Add -4026041 to 4028049.

$$a^{2}-4014a+4028049=2008$$

Factor a^{2}-4014a+4028049. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-2007\right)^{2}=2008$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2007\right)^{2}}=\sqrt{2008}$$

Simplify.

$$a-2007=2\sqrt{502} a-2007=-2\sqrt{502}$$

Add 2007 to both sides of the equation.

$$a=2\sqrt{502}+2007 a=2007-2\sqrt{502}$$