Solve (a+12)(a-4)=2a(a-4)

$$( a + 12 ) ( a - 4 ) = 2 a ( a - 4 )$$

Solve for a

$$a=12$$

$$a=4$$

Steps Using Factoring By Grouping

Use the distributive property to multiply a+12 by a-4 and combine like terms.

$$a^{2}+8a-48=2a\left(a-4\right)$$

Use the distributive property to multiply 2a by a-4.

$$a^{2}+8a-48=2a^{2}-8a$$

Subtract 2a^{2} from both sides.

$$a^{2}+8a-48-2a^{2}=-8a$$

Combine a^{2} and -2a^{2} to get -a^{2}.

$$-a^{2}+8a-48=-8a$$

Add 8a to both sides.

$$-a^{2}+8a-48+8a=0$$

Combine 8a and 8a to get 16a.

$$-a^{2}+16a-48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -a^{2}+aa+ba-48. To find a and b, set up a system to be solved.

$$a+b=16 ab=-\left(-48\right)=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 16.

$$a=12 b=4$$

Rewrite -a^{2}+16a-48 as \left(-a^{2}+12a\right)+\left(4a-48\right).

$$\left(-a^{2}+12a\right)+\left(4a-48\right)$$

Factor out -a in the first and 4 in the second group.

$$-a\left(a-12\right)+4\left(a-12\right)$$

Factor out common term a-12 by using distributive property.

$$\left(a-12\right)\left(-a+4\right)$$

To find equation solutions, solve a-12=0 and -a+4=0.

$$a=12 a=4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply a+12 by a-4 and combine like terms.

$$a^{2}+8a-48=2a\left(a-4\right)$$

Use the distributive property to multiply 2a by a-4.

$$a^{2}+8a-48=2a^{2}-8a$$

Subtract 2a^{2} from both sides.

$$a^{2}+8a-48-2a^{2}=-8a$$

Combine a^{2} and -2a^{2} to get -a^{2}.

$$-a^{2}+8a-48=-8a$$

Add 8a to both sides.

$$-a^{2}+8a-48+8a=0$$

Combine 8a and 8a to get 16a.

$$-a^{2}+16a-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 16 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-48\right)}}{2\left(-1\right)}$$

Square 16.

$$a=\frac{-16±\sqrt{256-4\left(-1\right)\left(-48\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-16±\sqrt{256+4\left(-48\right)}}{2\left(-1\right)}$$

Multiply 4 times -48.

$$a=\frac{-16±\sqrt{256-192}}{2\left(-1\right)}$$

Add 256 to -192.

$$a=\frac{-16±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$a=\frac{-16±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-16±8}{-2}$$

Now solve the equation a=\frac{-16±8}{-2} when ± is plus. Add -16 to 8.

$$a=-\frac{8}{-2}$$

Divide -8 by -2.

$$a=4$$

Now solve the equation a=\frac{-16±8}{-2} when ± is minus. Subtract 8 from -16.

$$a=-\frac{24}{-2}$$

Divide -24 by -2.

$$a=12$$

The equation is now solved.

$$a=4 a=12$$

Steps for Completing the Square

Use the distributive property to multiply a+12 by a-4 and combine like terms.

$$a^{2}+8a-48=2a\left(a-4\right)$$

Use the distributive property to multiply 2a by a-4.

$$a^{2}+8a-48=2a^{2}-8a$$

Subtract 2a^{2} from both sides.

$$a^{2}+8a-48-2a^{2}=-8a$$

Combine a^{2} and -2a^{2} to get -a^{2}.

$$-a^{2}+8a-48=-8a$$

Add 8a to both sides.

$$-a^{2}+8a-48+8a=0$$

Combine 8a and 8a to get 16a.

$$-a^{2}+16a-48=0$$

Add 48 to both sides. Anything plus zero gives itself.

$$-a^{2}+16a=48$$

Divide both sides by -1.

$$\frac{-a^{2}+16a}{-1}=\frac{48}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\frac{16}{-1}a=\frac{48}{-1}$$

Divide 16 by -1.

$$a^{2}-16a=\frac{48}{-1}$$

Divide 48 by -1.

$$a^{2}-16a=-48$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-16a+\left(-8\right)^{2}=-48+\left(-8\right)^{2}$$

Square -8.

$$a^{2}-16a+64=-48+64$$

Add -48 to 64.

$$a^{2}-16a+64=16$$

Factor a^{2}-16a+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-8\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-8\right)^{2}}=\sqrt{16}$$

Simplify.

$$a-8=4 a-8=-4$$

Add 8 to both sides of the equation.

$$a=12 a=4$$