Solve (12m^2-12m-6)^2-4(m^2)(36m^2-72m-36)=0

$$( 12 m ^ { 2 } - 12 m - 6 ) ^ { 2 } - 4 ( m ^ { 2 } ) ( 36 m ^ { 2 } - 72 m - 36 ) = 0$$

Solve for m

$$m=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

Square 12m^{2}-12m-6.

$$144m^{4}-288m^{3}+144m+36-4m^{2}\left(36m^{2}-72m-36\right)=0$$

Use the distributive property to multiply -4m^{2} by 36m^{2}-72m-36.

$$144m^{4}-288m^{3}+144m+36-144m^{4}+288m^{3}+144m^{2}=0$$

Combine 144m^{4} and -144m^{4} to get 0.

$$-288m^{3}+144m+36+288m^{3}+144m^{2}=0$$

Combine -288m^{3} and 288m^{3} to get 0.

$$144m+36+144m^{2}=0$$

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.

$$144m^{2}+144m+36=0$$

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

$$m=\frac{-144±\sqrt{144^{2}-4\times 144\times 36}}{2\times 144}$$

Square 144.

$$m=\frac{-144±\sqrt{20736-4\times 144\times 36}}{2\times 144}$$

Multiply -4 times 144.

$$m=\frac{-144±\sqrt{20736-576\times 36}}{2\times 144}$$

Multiply -576 times 36.

$$m=\frac{-144±\sqrt{20736-20736}}{2\times 144}$$

Add 20736 to -20736.

$$m=\frac{-144±\sqrt{0}}{2\times 144}$$

Take the square root of 0.

$$m=-\frac{144}{2\times 144}$$

Multiply 2 times 144.

$$m=-\frac{144}{288}$$

Reduce the fraction \frac{-144}{288} to lowest terms by extracting and canceling out 144.

$$m=-\frac{1}{2}$$

Steps for Completing the Square

Square 12m^{2}-12m-6.

$$144m^{4}-288m^{3}+144m+36-4m^{2}\left(36m^{2}-72m-36\right)=0$$

Use the distributive property to multiply -4m^{2} by 36m^{2}-72m-36.

$$144m^{4}-288m^{3}+144m+36-144m^{4}+288m^{3}+144m^{2}=0$$

Combine 144m^{4} and -144m^{4} to get 0.

$$-288m^{3}+144m+36+288m^{3}+144m^{2}=0$$

Combine -288m^{3} and 288m^{3} to get 0.

$$144m+36+144m^{2}=0$$

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$144m+144m^{2}=-36$$

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.

$$144m^{2}+144m=-36$$

Divide both sides by 144.

$$\frac{144m^{2}+144m}{144}=-\frac{36}{144}$$

Dividing by 144 undoes the multiplication by 144.

$$m^{2}+\frac{144}{144}m=-\frac{36}{144}$$

Divide 144 by 144.

$$m^{2}+m=-\frac{36}{144}$$

Reduce the fraction \frac{-36}{144} to lowest terms by extracting and canceling out 36.

$$m^{2}+m=-\frac{1}{4}$$

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

$$m^{2}+m+\left(\frac{1}{2}\right)^{2}=-\frac{1}{4}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+m+\frac{1}{4}=\frac{-1+1}{4}$$

Add -\frac{1}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+m+\frac{1}{4}=0$$

Factor m^{2}+m+\frac{1}{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}.

$$\left(m+\frac{1}{2}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{1}{2}\right)^{2}}=\sqrt{0}$$

Simplify.

$$m+\frac{1}{2}=0 m+\frac{1}{2}=0$$

Subtract \frac{1}{2} from both sides of the equation.

$$m=-\frac{1}{2} m=-\frac{1}{2}$$

The equation is now solved. Solutions are the same.

$$m=-\frac{1}{2}$$