Solve (x)=-2(x-3)(x-11)

$$( x ) = - 2 ( x - 3 ) ( x - 11 )$$

Solve for x

$$x = \frac{\sqrt{201} + 27}{4} \approx 10.29436172$$

$$x = \frac{27 - \sqrt{201}}{4} \approx 3.20563828$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2 by x-3.

$$x=\left(-2x+6\right)\left(x-11\right)$$

Use the distributive property to multiply -2x+6 by x-11 and combine like terms.

$$x=-2x^{2}+28x-66$$

Add 2x^{2} to both sides.

$$x+2x^{2}=28x-66$$

Subtract 28x from both sides.

$$x+2x^{2}-28x=-66$$

Combine x and -28x to get -27x.

$$-27x+2x^{2}=-66$$

Add 66 to both sides.

$$-27x+2x^{2}+66=0$$

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.

$$2x^{2}-27x+66=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -27 for b, and 66 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 2\times 66}}{2\times 2}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\times 2\times 66}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-27\right)±\sqrt{729-8\times 66}}{2\times 2}$$

Multiply -8 times 66.

$$x=\frac{-\left(-27\right)±\sqrt{729-528}}{2\times 2}$$

Add 729 to -528.

$$x=\frac{-\left(-27\right)±\sqrt{201}}{2\times 2}$$

The opposite of -27 is 27.

$$x=\frac{27±\sqrt{201}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{27±\sqrt{201}}{4}$$

Now solve the equation x=\frac{27±\sqrt{201}}{4} when ± is plus. Add 27 to \sqrt{201}.

$$x=\frac{\sqrt{201}+27}{4}$$

Now solve the equation x=\frac{27±\sqrt{201}}{4} when ± is minus. Subtract \sqrt{201} from 27.

$$x=\frac{27-\sqrt{201}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{201}+27}{4} x=\frac{27-\sqrt{201}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply -2 by x-3.

$$x=\left(-2x+6\right)\left(x-11\right)$$

Use the distributive property to multiply -2x+6 by x-11 and combine like terms.

$$x=-2x^{2}+28x-66$$

Add 2x^{2} to both sides.

$$x+2x^{2}=28x-66$$

Subtract 28x from both sides.

$$x+2x^{2}-28x=-66$$

Combine x and -28x to get -27x.

$$-27x+2x^{2}=-66$$

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.

$$2x^{2}-27x=-66$$

Divide both sides by 2.

$$\frac{2x^{2}-27x}{2}=-\frac{66}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{27}{2}x=-\frac{66}{2}$$

Divide -66 by 2.

$$x^{2}-\frac{27}{2}x=-33$$

Divide -\frac{27}{2}, the coefficient of the x term, by 2 to get -\frac{27}{4}. Then add the square of -\frac{27}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{27}{2}x+\left(-\frac{27}{4}\right)^{2}=-33+\left(-\frac{27}{4}\right)^{2}$$

Square -\frac{27}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{27}{2}x+\frac{729}{16}=-33+\frac{729}{16}$$

Add -33 to \frac{729}{16}.

$$x^{2}-\frac{27}{2}x+\frac{729}{16}=\frac{201}{16}$$

Factor x^{2}-\frac{27}{2}x+\frac{729}{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}.

$$\left(x-\frac{27}{4}\right)^{2}=\frac{201}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{4}\right)^{2}}=\sqrt{\frac{201}{16}}$$

Simplify.

$$x-\frac{27}{4}=\frac{\sqrt{201}}{4} x-\frac{27}{4}=-\frac{\sqrt{201}}{4}$$

Add \frac{27}{4} to both sides of the equation.

$$x=\frac{\sqrt{201}+27}{4} x=\frac{27-\sqrt{201}}{4}$$