Solve (6x-11)^2=12x

$$( 6 x - 11 ) ^ { 2 } = 12 x$$

Solve for x

$$x=\frac{\sqrt{23}}{6}+2\approx 2.799305254$$

$$x=-\frac{\sqrt{23}}{6}+2\approx 1.200694746$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-11\right)^{2}.

$$36x^{2}-132x+121=12x$$

Subtract 12x from both sides.

$$36x^{2}-132x+121-12x=0$$

Combine -132x and -12x to get -144x.

$$36x^{2}-144x+121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -144 for b, and 121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}-4\times 36\times 121}}{2\times 36}$$

Square -144.

$$x=\frac{-\left(-144\right)±\sqrt{20736-4\times 36\times 121}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-144\right)±\sqrt{20736-144\times 121}}{2\times 36}$$

Multiply -144 times 121.

$$x=\frac{-\left(-144\right)±\sqrt{20736-17424}}{2\times 36}$$

Add 20736 to -17424.

$$x=\frac{-\left(-144\right)±\sqrt{3312}}{2\times 36}$$

Take the square root of 3312.

$$x=\frac{-\left(-144\right)±12\sqrt{23}}{2\times 36}$$

The opposite of -144 is 144.

$$x=\frac{144±12\sqrt{23}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{144±12\sqrt{23}}{72}$$

Now solve the equation x=\frac{144±12\sqrt{23}}{72} when ± is plus. Add 144 to 12\sqrt{23}.

$$x=\frac{12\sqrt{23}+144}{72}$$

Divide 144+12\sqrt{23} by 72.

$$x=\frac{\sqrt{23}}{6}+2$$

Now solve the equation x=\frac{144±12\sqrt{23}}{72} when ± is minus. Subtract 12\sqrt{23} from 144.

$$x=\frac{144-12\sqrt{23}}{72}$$

Divide 144-12\sqrt{23} by 72.

$$x=-\frac{\sqrt{23}}{6}+2$$

The equation is now solved.

$$x=\frac{\sqrt{23}}{6}+2 x=-\frac{\sqrt{23}}{6}+2$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-11\right)^{2}.

$$36x^{2}-132x+121=12x$$

Subtract 12x from both sides.

$$36x^{2}-132x+121-12x=0$$

Combine -132x and -12x to get -144x.

$$36x^{2}-144x+121=0$$

Subtract 121 from both sides. Anything subtracted from zero gives its negation.

$$36x^{2}-144x=-121$$

Divide both sides by 36.

$$\frac{36x^{2}-144x}{36}=-\frac{121}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{144}{36}\right)x=-\frac{121}{36}$$

Divide -144 by 36.

$$x^{2}-4x=-\frac{121}{36}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{121}{36}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{121}{36}+4$$

Add -\frac{121}{36} to 4.

$$x^{2}-4x+4=\frac{23}{36}$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=\frac{23}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{23}{36}}$$

Simplify.

$$x-2=\frac{\sqrt{23}}{6} x-2=-\frac{\sqrt{23}}{6}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{23}}{6}+2 x=-\frac{\sqrt{23}}{6}+2$$