Solve x^2=(6x+11)^2

$$x ^ { 2 } = ( 6 x + 11 ) ^ { 2 }$$

Solve for x

$$x = -\frac{11}{7} = -1\frac{4}{7} \approx -1.571428571$$

$$x = -\frac{11}{5} = -2\frac{1}{5} = -2.2$$

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}.

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

Subtract 36x^{2} from both sides.

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

Combine x^{2} and -36x^{2} to get -35x^{2}.

$$-35x^{2}=132x+121$$

Subtract 132x from both sides.

$$-35x^{2}-132x=121$$

Subtract 121 from both sides.

$$-35x^{2}-132x-121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -35 for a, -132 for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-132\right)±\sqrt{\left(-132\right)^{2}-4\left(-35\right)\left(-121\right)}}{2\left(-35\right)}$$

Square -132.

$$x=\frac{-\left(-132\right)±\sqrt{17424-4\left(-35\right)\left(-121\right)}}{2\left(-35\right)}$$

Multiply -4 times -35.

$$x=\frac{-\left(-132\right)±\sqrt{17424+140\left(-121\right)}}{2\left(-35\right)}$$

Multiply 140 times -121.

$$x=\frac{-\left(-132\right)±\sqrt{17424-16940}}{2\left(-35\right)}$$

Add 17424 to -16940.

$$x=\frac{-\left(-132\right)±\sqrt{484}}{2\left(-35\right)}$$

Take the square root of 484.

$$x=\frac{-\left(-132\right)±22}{2\left(-35\right)}$$

The opposite of -132 is 132.

$$x=\frac{132±22}{2\left(-35\right)}$$

Multiply 2 times -35.

$$x=\frac{132±22}{-70}$$

Now solve the equation x=\frac{132±22}{-70} when ± is plus. Add 132 to 22.

$$x=\frac{154}{-70}$$

Reduce the fraction \frac{154}{-70} to lowest terms by extracting and canceling out 14.

$$x=-\frac{11}{5}$$

Now solve the equation x=\frac{132±22}{-70} when ± is minus. Subtract 22 from 132.

$$x=\frac{110}{-70}$$

Reduce the fraction \frac{110}{-70} to lowest terms by extracting and canceling out 10.

$$x=-\frac{11}{7}$$

The equation is now solved.

$$x=-\frac{11}{5} x=-\frac{11}{7}$$

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}.

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

Subtract 36x^{2} from both sides.

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

Combine x^{2} and -36x^{2} to get -35x^{2}.

$$-35x^{2}=132x+121$$

Subtract 132x from both sides.

$$-35x^{2}-132x=121$$

Divide both sides by -35.

$$\frac{-35x^{2}-132x}{-35}=\frac{121}{-35}$$

Dividing by -35 undoes the multiplication by -35.

$$x^{2}+\left(-\frac{132}{-35}\right)x=\frac{121}{-35}$$

Divide -132 by -35.

$$x^{2}+\frac{132}{35}x=\frac{121}{-35}$$

Divide 121 by -35.

$$x^{2}+\frac{132}{35}x=-\frac{121}{35}$$

Divide \frac{132}{35}, the coefficient of the x term, by 2 to get \frac{66}{35}. Then add the square of \frac{66}{35} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{132}{35}x+\left(\frac{66}{35}\right)^{2}=-\frac{121}{35}+\left(\frac{66}{35}\right)^{2}$$

Square \frac{66}{35} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{132}{35}x+\frac{4356}{1225}=-\frac{121}{35}+\frac{4356}{1225}$$

Add -\frac{121}{35} to \frac{4356}{1225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{132}{35}x+\frac{4356}{1225}=\frac{121}{1225}$$

Factor x^{2}+\frac{132}{35}x+\frac{4356}{1225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{66}{35}\right)^{2}=\frac{121}{1225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{66}{35}\right)^{2}}=\sqrt{\frac{121}{1225}}$$

Simplify.

$$x+\frac{66}{35}=\frac{11}{35} x+\frac{66}{35}=-\frac{11}{35}$$

Subtract \frac{66}{35} from both sides of the equation.

$$x=-\frac{11}{7} x=-\frac{11}{5}$$