Solve (8-2x)(5-2x)=6

$$( 8 - 2 x ) ( 5 - 2 x ) = 6$$

Solve for x

$$x = \frac{13 - \sqrt{33}}{4} \approx 1.813859338$$

$$x = \frac{\sqrt{33} + 13}{4} \approx 4.686140662$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8-2x by 5-2x and combine like terms.

$$40-26x+4x^{2}=6$$

Subtract 6 from both sides.

$$40-26x+4x^{2}-6=0$$

Subtract 6 from 40 to get 34.

$$34-26x+4x^{2}=0$$

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.

$$4x^{2}-26x+34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -26 for b, and 34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 4\times 34}}{2\times 4}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 4\times 34}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-26\right)±\sqrt{676-16\times 34}}{2\times 4}$$

Multiply -16 times 34.

$$x=\frac{-\left(-26\right)±\sqrt{676-544}}{2\times 4}$$

Add 676 to -544.

$$x=\frac{-\left(-26\right)±\sqrt{132}}{2\times 4}$$

Take the square root of 132.

$$x=\frac{-\left(-26\right)±2\sqrt{33}}{2\times 4}$$

The opposite of -26 is 26.

$$x=\frac{26±2\sqrt{33}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{26±2\sqrt{33}}{8}$$

Now solve the equation x=\frac{26±2\sqrt{33}}{8} when ± is plus. Add 26 to 2\sqrt{33}.

$$x=\frac{2\sqrt{33}+26}{8}$$

Divide 26+2\sqrt{33} by 8.

$$x=\frac{\sqrt{33}+13}{4}$$

Now solve the equation x=\frac{26±2\sqrt{33}}{8} when ± is minus. Subtract 2\sqrt{33} from 26.

$$x=\frac{26-2\sqrt{33}}{8}$$

Divide 26-2\sqrt{33} by 8.

$$x=\frac{13-\sqrt{33}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{33}+13}{4} x=\frac{13-\sqrt{33}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 8-2x by 5-2x and combine like terms.

$$40-26x+4x^{2}=6$$

Subtract 40 from both sides.

$$-26x+4x^{2}=6-40$$

Subtract 40 from 6 to get -34.

$$-26x+4x^{2}=-34$$

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.

$$4x^{2}-26x=-34$$

Divide both sides by 4.

$$\frac{4x^{2}-26x}{4}=-\frac{34}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{26}{4}\right)x=-\frac{34}{4}$$

Reduce the fraction \frac{-26}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{13}{2}x=-\frac{34}{4}$$

Reduce the fraction \frac{-34}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{13}{2}x=-\frac{17}{2}$$

Divide -\frac{13}{2}, the coefficient of the x term, by 2 to get -\frac{13}{4}. Then add the square of -\frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-\frac{17}{2}+\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{17}{2}+\frac{169}{16}$$

Add -\frac{17}{2} to \frac{169}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{33}{16}$$

Factor x^{2}-\frac{13}{2}x+\frac{169}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{33}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{33}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{\sqrt{33}}{4} x-\frac{13}{4}=-\frac{\sqrt{33}}{4}$$

Add \frac{13}{4} to both sides of the equation.

$$x=\frac{\sqrt{33}+13}{4} x=\frac{13-\sqrt{33}}{4}$$