Solve 4m^2-16=12(m-2)^2

$$4 m ^ { 2 } - 16 = 12 ( m - 2 ) ^ { 2 }$$

Solve for m

$$m=2$$

$$m=4$$

Steps Using Factoring By Grouping

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

$$4m^{2}-16=12\left(m^{2}-4m+4\right)$$

Use the distributive property to multiply 12 by m^{2}-4m+4.

$$4m^{2}-16=12m^{2}-48m+48$$

Subtract 12m^{2} from both sides.

$$4m^{2}-16-12m^{2}=-48m+48$$

Combine 4m^{2} and -12m^{2} to get -8m^{2}.

$$-8m^{2}-16=-48m+48$$

Add 48m to both sides.

$$-8m^{2}-16+48m=48$$

Subtract 48 from both sides.

$$-8m^{2}-16+48m-48=0$$

Subtract 48 from -16 to get -64.

$$-8m^{2}-64+48m=0$$

Divide both sides by 8.

$$-m^{2}-8+6m=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-m^{2}+6m-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -m^{2}+am+bm-8. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-8\right)=8$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 6.

$$a=4 b=2$$

Rewrite -m^{2}+6m-8 as \left(-m^{2}+4m\right)+\left(2m-8\right).

$$\left(-m^{2}+4m\right)+\left(2m-8\right)$$

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

$$-m\left(m-4\right)+2\left(m-4\right)$$

Factor out common term m-4 by using distributive property.

$$\left(m-4\right)\left(-m+2\right)$$

To find equation solutions, solve m-4=0 and -m+2=0.

$$m=4 m=2$$

Steps Using the Quadratic Formula

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

$$4m^{2}-16=12\left(m^{2}-4m+4\right)$$

Use the distributive property to multiply 12 by m^{2}-4m+4.

$$4m^{2}-16=12m^{2}-48m+48$$

Subtract 12m^{2} from both sides.

$$4m^{2}-16-12m^{2}=-48m+48$$

Combine 4m^{2} and -12m^{2} to get -8m^{2}.

$$-8m^{2}-16=-48m+48$$

Add 48m to both sides.

$$-8m^{2}-16+48m=48$$

Subtract 48 from both sides.

$$-8m^{2}-16+48m-48=0$$

Subtract 48 from -16 to get -64.

$$-8m^{2}-64+48m=0$$

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.

$$-8m^{2}+48m-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 48 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-48±\sqrt{48^{2}-4\left(-8\right)\left(-64\right)}}{2\left(-8\right)}$$

Square 48.

$$m=\frac{-48±\sqrt{2304-4\left(-8\right)\left(-64\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$m=\frac{-48±\sqrt{2304+32\left(-64\right)}}{2\left(-8\right)}$$

Multiply 32 times -64.

$$m=\frac{-48±\sqrt{2304-2048}}{2\left(-8\right)}$$

Add 2304 to -2048.

$$m=\frac{-48±\sqrt{256}}{2\left(-8\right)}$$

Take the square root of 256.

$$m=\frac{-48±16}{2\left(-8\right)}$$

Multiply 2 times -8.

$$m=\frac{-48±16}{-16}$$

Now solve the equation m=\frac{-48±16}{-16} when ± is plus. Add -48 to 16.

$$m=-\frac{32}{-16}$$

Divide -32 by -16.

$$m=2$$

Now solve the equation m=\frac{-48±16}{-16} when ± is minus. Subtract 16 from -48.

$$m=-\frac{64}{-16}$$

Divide -64 by -16.

$$m=4$$

The equation is now solved.

$$m=2 m=4$$

Steps for Completing the Square

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

$$4m^{2}-16=12\left(m^{2}-4m+4\right)$$

Use the distributive property to multiply 12 by m^{2}-4m+4.

$$4m^{2}-16=12m^{2}-48m+48$$

Subtract 12m^{2} from both sides.

$$4m^{2}-16-12m^{2}=-48m+48$$

Combine 4m^{2} and -12m^{2} to get -8m^{2}.

$$-8m^{2}-16=-48m+48$$

Add 48m to both sides.

$$-8m^{2}-16+48m=48$$

Add 16 to both sides.

$$-8m^{2}+48m=48+16$$

Add 48 and 16 to get 64.

$$-8m^{2}+48m=64$$

Divide both sides by -8.

$$\frac{-8m^{2}+48m}{-8}=\frac{64}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$m^{2}+\frac{48}{-8}m=\frac{64}{-8}$$

Divide 48 by -8.

$$m^{2}-6m=\frac{64}{-8}$$

Divide 64 by -8.

$$m^{2}-6m=-8$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-6m+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Square -3.

$$m^{2}-6m+9=-8+9$$

Add -8 to 9.

$$m^{2}-6m+9=1$$

Factor m^{2}-6m+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-3\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-3\right)^{2}}=\sqrt{1}$$

Simplify.

$$m-3=1 m-3=-1$$

Add 3 to both sides of the equation.

$$m=4 m=2$$