Solve 2(m-6)(m-4)/(m-2)^2=6

$$\frac { 2 ( m - 6 ) ( m - 4 ) } { ( m - 2 ) ^ { 2 } } = 6$$

Solve for m

$$m=-2$$

$$m=3$$

Steps Using Factoring By Grouping

Variable m cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by \left(m-2\right)^{2}.

$$2\left(m-6\right)\left(m-4\right)=6\left(m-2\right)^{2}$$

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

$$\left(2m-12\right)\left(m-4\right)=6\left(m-2\right)^{2}$$

Use the distributive property to multiply 2m-12 by m-4 and combine like terms.

$$2m^{2}-20m+48=6\left(m-2\right)^{2}$$

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

$$2m^{2}-20m+48=6\left(m^{2}-4m+4\right)$$

Use the distributive property to multiply 6 by m^{2}-4m+4.

$$2m^{2}-20m+48=6m^{2}-24m+24$$

Subtract 6m^{2} from both sides.

$$2m^{2}-20m+48-6m^{2}=-24m+24$$

Combine 2m^{2} and -6m^{2} to get -4m^{2}.

$$-4m^{2}-20m+48=-24m+24$$

Add 24m to both sides.

$$-4m^{2}-20m+48+24m=24$$

Combine -20m and 24m to get 4m.

$$-4m^{2}+4m+48=24$$

Subtract 24 from both sides.

$$-4m^{2}+4m+48-24=0$$

Subtract 24 from 48 to get 24.

$$-4m^{2}+4m+24=0$$

Divide both sides by 4.

$$-m^{2}+m+6=0$$

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

$$a+b=1 ab=-6=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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 -m^{2}+m+6 as \left(-m^{2}+3m\right)+\left(-2m+6\right).

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

Factor out -m in the first and -2 in the second group.

$$-m\left(m-3\right)-2\left(m-3\right)$$

Factor out common term m-3 by using distributive property.

$$\left(m-3\right)\left(-m-2\right)$$

To find equation solutions, solve m-3=0 and -m-2=0.

$$m=3 m=-2$$

Steps Using the Quadratic Formula

Variable m cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by \left(m-2\right)^{2}.

$$2\left(m-6\right)\left(m-4\right)=6\left(m-2\right)^{2}$$

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

$$\left(2m-12\right)\left(m-4\right)=6\left(m-2\right)^{2}$$

Use the distributive property to multiply 2m-12 by m-4 and combine like terms.

$$2m^{2}-20m+48=6\left(m-2\right)^{2}$$

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

$$2m^{2}-20m+48=6\left(m^{2}-4m+4\right)$$

Use the distributive property to multiply 6 by m^{2}-4m+4.

$$2m^{2}-20m+48=6m^{2}-24m+24$$

Subtract 6m^{2} from both sides.

$$2m^{2}-20m+48-6m^{2}=-24m+24$$

Combine 2m^{2} and -6m^{2} to get -4m^{2}.

$$-4m^{2}-20m+48=-24m+24$$

Add 24m to both sides.

$$-4m^{2}-20m+48+24m=24$$

Combine -20m and 24m to get 4m.

$$-4m^{2}+4m+48=24$$

Subtract 24 from both sides.

$$-4m^{2}+4m+48-24=0$$

Subtract 24 from 48 to get 24.

$$-4m^{2}+4m+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 4 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-4±\sqrt{4^{2}-4\left(-4\right)\times 24}}{2\left(-4\right)}$$

Square 4.

$$m=\frac{-4±\sqrt{16-4\left(-4\right)\times 24}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$m=\frac{-4±\sqrt{16+16\times 24}}{2\left(-4\right)}$$

Multiply 16 times 24.

$$m=\frac{-4±\sqrt{16+384}}{2\left(-4\right)}$$

Add 16 to 384.

$$m=\frac{-4±\sqrt{400}}{2\left(-4\right)}$$

Take the square root of 400.

$$m=\frac{-4±20}{2\left(-4\right)}$$

Multiply 2 times -4.

$$m=\frac{-4±20}{-8}$$

Now solve the equation m=\frac{-4±20}{-8} when ± is plus. Add -4 to 20.

$$m=\frac{16}{-8}$$

Divide 16 by -8.

$$m=-2$$

Now solve the equation m=\frac{-4±20}{-8} when ± is minus. Subtract 20 from -4.

$$m=-\frac{24}{-8}$$

Divide -24 by -8.

$$m=3$$

The equation is now solved.

$$m=-2 m=3$$

Steps for Completing the Square

Variable m cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by \left(m-2\right)^{2}.

$$2\left(m-6\right)\left(m-4\right)=6\left(m-2\right)^{2}$$

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

$$\left(2m-12\right)\left(m-4\right)=6\left(m-2\right)^{2}$$

Use the distributive property to multiply 2m-12 by m-4 and combine like terms.

$$2m^{2}-20m+48=6\left(m-2\right)^{2}$$

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

$$2m^{2}-20m+48=6\left(m^{2}-4m+4\right)$$

Use the distributive property to multiply 6 by m^{2}-4m+4.

$$2m^{2}-20m+48=6m^{2}-24m+24$$

Subtract 6m^{2} from both sides.

$$2m^{2}-20m+48-6m^{2}=-24m+24$$

Combine 2m^{2} and -6m^{2} to get -4m^{2}.

$$-4m^{2}-20m+48=-24m+24$$

Add 24m to both sides.

$$-4m^{2}-20m+48+24m=24$$

Combine -20m and 24m to get 4m.

$$-4m^{2}+4m+48=24$$

Subtract 48 from both sides.

$$-4m^{2}+4m=24-48$$

Subtract 48 from 24 to get -24.

$$-4m^{2}+4m=-24$$

Divide both sides by -4.

$$\frac{-4m^{2}+4m}{-4}=-\frac{24}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$m^{2}+\frac{4}{-4}m=-\frac{24}{-4}$$

Divide 4 by -4.

$$m^{2}-m=-\frac{24}{-4}$$

Divide -24 by -4.

$$m^{2}-m=6$$

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.

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

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

$$m^{2}-m+\frac{1}{4}=6+\frac{1}{4}$$

Add 6 to \frac{1}{4}.

$$m^{2}-m+\frac{1}{4}=\frac{25}{4}$$

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}.

$$\left(m-\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$m-\frac{1}{2}=\frac{5}{2} m-\frac{1}{2}=-\frac{5}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$m=3 m=-2$$