Solve (6-x)(2x-5)+30=0

$$(6-x)(2x-5)+30=0$$

Solve for x

$$x = \frac{17}{2} = 8\frac{1}{2} = 8.5$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 6-x by 2x-5 and combine like terms.

$$17x-30-2x^{2}+30=0$$

Add -30 and 30 to get 0.

$$17x-2x^{2}=0$$

Factor out x.

$$x\left(17-2x\right)=0$$

To find equation solutions, solve x=0 and 17-2x=0.

$$x=0 x=\frac{17}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6-x by 2x-5 and combine like terms.

$$17x-30-2x^{2}+30=0$$

Add -30 and 30 to get 0.

$$17x-2x^{2}=0$$

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.

$$-2x^{2}+17x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 17 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of 17^{2}.

$$x=\frac{-17±17}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-17±17}{-4}$$

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

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

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

$$x=-\frac{34}{-4}$$

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

$$x=\frac{17}{2}$$

The equation is now solved.

$$x=0 x=\frac{17}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 6-x by 2x-5 and combine like terms.

$$17x-30-2x^{2}+30=0$$

Add -30 and 30 to get 0.

$$17x-2x^{2}=0$$

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.

$$-2x^{2}+17x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}+17x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{17}{-2}x=\frac{0}{-2}$$

Divide 17 by -2.

$$x^{2}-\frac{17}{2}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}-\frac{17}{2}x=0$$

Divide -\frac{17}{2}, the coefficient of the x term, by 2 to get -\frac{17}{4}. Then add the square of -\frac{17}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{17}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{2}x+\frac{289}{16}=\frac{289}{16}$$

Factor x^{2}-\frac{17}{2}x+\frac{289}{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}.

$$\left(x-\frac{17}{4}\right)^{2}=\frac{289}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{4}\right)^{2}}=\sqrt{\frac{289}{16}}$$

Simplify.

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

Add \frac{17}{4} to both sides of the equation.

$$x=\frac{17}{2} x=0$$