Solve 2x(6-x)=1

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

Solve for x

$$x=\frac{\sqrt{34}}{2}+3\approx 5.915475947$$

$$x=-\frac{\sqrt{34}}{2}+3\approx 0.084524053$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 6-x.

$$12x-2x^{2}=1$$

Subtract 1 from both sides.

$$12x-2x^{2}-1=0$$

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.

$$-2x^{2}+12x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 12 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-2\right)\left(-1\right)}}{2\left(-2\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-2\right)\left(-1\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-12±\sqrt{144+8\left(-1\right)}}{2\left(-2\right)}$$

Multiply 8 times -1.

$$x=\frac{-12±\sqrt{144-8}}{2\left(-2\right)}$$

Add 144 to -8.

$$x=\frac{-12±\sqrt{136}}{2\left(-2\right)}$$

Take the square root of 136.

$$x=\frac{-12±2\sqrt{34}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-12±2\sqrt{34}}{-4}$$

Now solve the equation x=\frac{-12±2\sqrt{34}}{-4} when ± is plus. Add -12 to 2\sqrt{34}.

$$x=\frac{2\sqrt{34}-12}{-4}$$

Divide -12+2\sqrt{34} by -4.

$$x=-\frac{\sqrt{34}}{2}+3$$

Now solve the equation x=\frac{-12±2\sqrt{34}}{-4} when ± is minus. Subtract 2\sqrt{34} from -12.

$$x=\frac{-2\sqrt{34}-12}{-4}$$

Divide -12-2\sqrt{34} by -4.

$$x=\frac{\sqrt{34}}{2}+3$$

The equation is now solved.

$$x=-\frac{\sqrt{34}}{2}+3 x=\frac{\sqrt{34}}{2}+3$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 6-x.

$$12x-2x^{2}=1$$

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.

$$-2x^{2}+12x=1$$

Divide both sides by -2.

$$\frac{-2x^{2}+12x}{-2}=\frac{1}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{12}{-2}x=\frac{1}{-2}$$

Divide 12 by -2.

$$x^{2}-6x=\frac{1}{-2}$$

Divide 1 by -2.

$$x^{2}-6x=-\frac{1}{2}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-\frac{1}{2}+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-\frac{1}{2}+9$$

Add -\frac{1}{2} to 9.

$$x^{2}-6x+9=\frac{17}{2}$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=\frac{17}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{\frac{17}{2}}$$

Simplify.

$$x-3=\frac{\sqrt{34}}{2} x-3=-\frac{\sqrt{34}}{2}$$

Add 3 to both sides of the equation.

$$x=\frac{\sqrt{34}}{2}+3 x=-\frac{\sqrt{34}}{2}+3$$