Solve 1-2(x-3)(x-11)=0

$$1 - 2 ( x - 3 ) ( x - 11 ) = 0$$

Solve for x

$$x=\frac{\sqrt{66}}{2}+7\approx 11.062019202$$

$$x=-\frac{\sqrt{66}}{2}+7\approx 2.937980798$$

Steps Using the Quadratic Formula

Multiply -1 and 2 to get -2.

$$1-2\left(x-3\right)\left(x-11\right)=0$$

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

$$1+\left(-2x+6\right)\left(x-11\right)=0$$

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

$$1-2x^{2}+28x-66=0$$

Subtract 66 from 1 to get -65.

$$-65-2x^{2}+28x=0$$

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.

$$-2x^{2}+28x-65=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 28 for b, and -65 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-2\right)\left(-65\right)}}{2\left(-2\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-2\right)\left(-65\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-28±\sqrt{784+8\left(-65\right)}}{2\left(-2\right)}$$

Multiply 8 times -65.

$$x=\frac{-28±\sqrt{784-520}}{2\left(-2\right)}$$

Add 784 to -520.

$$x=\frac{-28±\sqrt{264}}{2\left(-2\right)}$$

Take the square root of 264.

$$x=\frac{-28±2\sqrt{66}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-28±2\sqrt{66}}{-4}$$

Now solve the equation x=\frac{-28±2\sqrt{66}}{-4} when ± is plus. Add -28 to 2\sqrt{66}.

$$x=\frac{2\sqrt{66}-28}{-4}$$

Divide -28+2\sqrt{66} by -4.

$$x=-\frac{\sqrt{66}}{2}+7$$

Now solve the equation x=\frac{-28±2\sqrt{66}}{-4} when ± is minus. Subtract 2\sqrt{66} from -28.

$$x=\frac{-2\sqrt{66}-28}{-4}$$

Divide -28-2\sqrt{66} by -4.

$$x=\frac{\sqrt{66}}{2}+7$$

The equation is now solved.

$$x=-\frac{\sqrt{66}}{2}+7 x=\frac{\sqrt{66}}{2}+7$$

Steps for Completing the Square

Multiply -1 and 2 to get -2.

$$1-2\left(x-3\right)\left(x-11\right)=0$$

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

$$1+\left(-2x+6\right)\left(x-11\right)=0$$

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

$$1-2x^{2}+28x-66=0$$

Subtract 66 from 1 to get -65.

$$-65-2x^{2}+28x=0$$

Add 65 to both sides. Anything plus zero gives itself.

$$-2x^{2}+28x=65$$

Divide both sides by -2.

$$\frac{-2x^{2}+28x}{-2}=\frac{65}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{28}{-2}x=\frac{65}{-2}$$

Divide 28 by -2.

$$x^{2}-14x=\frac{65}{-2}$$

Divide 65 by -2.

$$x^{2}-14x=-\frac{65}{2}$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-\frac{65}{2}+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-\frac{65}{2}+49$$

Add -\frac{65}{2} to 49.

$$x^{2}-14x+49=\frac{33}{2}$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=\frac{33}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{\frac{33}{2}}$$

Simplify.

$$x-7=\frac{\sqrt{66}}{2} x-7=-\frac{\sqrt{66}}{2}$$

Add 7 to both sides of the equation.

$$x=\frac{\sqrt{66}}{2}+7 x=-\frac{\sqrt{66}}{2}+7$$