Solve left(6xright)^2-6x-6=0

$${ \left(6x \right) }^{ 2 } -6x-6 = 0$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x=\frac{1}{2}=0.5$$

Steps Using Factoring By Grouping

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}-6x-6=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}-6x-6=0$$

Divide both sides by 6.

$$6x^{2}-x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-1 ab=6\left(-1\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -1.

$$a=-3 b=2$$

Rewrite 6x^{2}-x-1 as \left(6x^{2}-3x\right)+\left(2x-1\right).

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

Factor out 3x in 6x^{2}-3x.

$$3x\left(2x-1\right)+2x-1$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(3x+1\right)$$

To find equation solutions, solve 2x-1=0 and 3x+1=0.

$$x=\frac{1}{2} x=-\frac{1}{3}$$

Steps Using the Quadratic Formula

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}-6x-6=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}-6x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -6 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 36\left(-6\right)}}{2\times 36}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 36\left(-6\right)}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-6\right)±\sqrt{36-144\left(-6\right)}}{2\times 36}$$

Multiply -144 times -6.

$$x=\frac{-\left(-6\right)±\sqrt{36+864}}{2\times 36}$$

Add 36 to 864.

$$x=\frac{-\left(-6\right)±\sqrt{900}}{2\times 36}$$

Take the square root of 900.

$$x=\frac{-\left(-6\right)±30}{2\times 36}$$

The opposite of -6 is 6.

$$x=\frac{6±30}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{6±30}{72}$$

Now solve the equation x=\frac{6±30}{72} when ± is plus. Add 6 to 30.

$$x=\frac{36}{72}$$

Reduce the fraction \frac{36}{72} to lowest terms by extracting and canceling out 36.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{6±30}{72} when ± is minus. Subtract 30 from 6.

$$x=-\frac{24}{72}$$

Reduce the fraction \frac{-24}{72} to lowest terms by extracting and canceling out 24.

$$x=-\frac{1}{3}$$

The equation is now solved.

$$x=\frac{1}{2} x=-\frac{1}{3}$$

Steps for Completing the Square

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}-6x-6=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}-6x-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$36x^{2}-6x=6$$

Divide both sides by 36.

$$\frac{36x^{2}-6x}{36}=\frac{6}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{6}{36}\right)x=\frac{6}{36}$$

Reduce the fraction \frac{-6}{36} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{1}{6}x=\frac{6}{36}$$

Reduce the fraction \frac{6}{36} to lowest terms by extracting and canceling out 6.

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

Divide -\frac{1}{6}, the coefficient of the x term, by 2 to get -\frac{1}{12}. Then add the square of -\frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=\frac{1}{6}+\left(-\frac{1}{12}\right)^{2}$$

Square -\frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{1}{6}+\frac{1}{144}$$

Add \frac{1}{6} to \frac{1}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{25}{144}$$

Factor x^{2}-\frac{1}{6}x+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{25}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{25}{144}}$$

Simplify.

$$x-\frac{1}{12}=\frac{5}{12} x-\frac{1}{12}=-\frac{5}{12}$$

Add \frac{1}{12} to both sides of the equation.

$$x=\frac{1}{2} x=-\frac{1}{3}$$