Solve (6-x)(2x-1)=13

$$( 6 - x ) ( 2 x - 1 ) = 13$$

Solve for x

$$x = \frac{\sqrt{17} + 13}{4} \approx 4.280776406$$

$$x = \frac{13 - \sqrt{17}}{4} \approx 2.219223594$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6-x by 2x-1 and combine like terms.

$$13x-6-2x^{2}=13$$

Subtract 13 from both sides.

$$13x-6-2x^{2}-13=0$$

Subtract 13 from -6 to get -19.

$$13x-19-2x^{2}=0$$

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.

$$-2x^{2}+13x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 13 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-19\right)}}{2\left(-2\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-19\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-13±\sqrt{169+8\left(-19\right)}}{2\left(-2\right)}$$

Multiply 8 times -19.

$$x=\frac{-13±\sqrt{169-152}}{2\left(-2\right)}$$

Add 169 to -152.

$$x=\frac{-13±\sqrt{17}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-13±\sqrt{17}}{-4}$$

Now solve the equation x=\frac{-13±\sqrt{17}}{-4} when ± is plus. Add -13 to \sqrt{17}.

$$x=\frac{\sqrt{17}-13}{-4}$$

Divide -13+\sqrt{17} by -4.

$$x=\frac{13-\sqrt{17}}{4}$$

Now solve the equation x=\frac{-13±\sqrt{17}}{-4} when ± is minus. Subtract \sqrt{17} from -13.

$$x=\frac{-\sqrt{17}-13}{-4}$$

Divide -13-\sqrt{17} by -4.

$$x=\frac{\sqrt{17}+13}{4}$$

The equation is now solved.

$$x=\frac{13-\sqrt{17}}{4} x=\frac{\sqrt{17}+13}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 6-x by 2x-1 and combine like terms.

$$13x-6-2x^{2}=13$$

Add 6 to both sides.

$$13x-2x^{2}=13+6$$

Add 13 and 6 to get 19.

$$13x-2x^{2}=19$$

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.

$$-2x^{2}+13x=19$$

Divide both sides by -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{19}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{13}{-2}x=\frac{19}{-2}$$

Divide 13 by -2.

$$x^{2}-\frac{13}{2}x=\frac{19}{-2}$$

Divide 19 by -2.

$$x^{2}-\frac{13}{2}x=-\frac{19}{2}$$

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.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-\frac{19}{2}+\left(-\frac{13}{4}\right)^{2}$$

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

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{19}{2}+\frac{169}{16}$$

Add -\frac{19}{2} to \frac{169}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{17}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{\sqrt{17}}{4} x-\frac{13}{4}=-\frac{\sqrt{17}}{4}$$

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

$$x=\frac{\sqrt{17}+13}{4} x=\frac{13-\sqrt{17}}{4}$$