Solve x^2-(x-12)^2=(x-24)^2

$$x ^ { 2 } - ( x - 12 ) ^ { 2 } = ( x - 24 ) ^ { 2 }$$

Solve for x

$$x=12$$

$$x=60$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$x^{2}-\left(x^{2}-24x+144\right)=\left(x-24\right)^{2}$$

To find the opposite of x^{2}-24x+144, find the opposite of each term.

$$x^{2}-x^{2}+24x-144=\left(x-24\right)^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$24x-144=\left(x-24\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-24\right)^{2}.

$$24x-144=x^{2}-48x+576$$

Subtract x^{2} from both sides.

$$24x-144-x^{2}=-48x+576$$

Add 48x to both sides.

$$24x-144-x^{2}+48x=576$$

Combine 24x and 48x to get 72x.

$$72x-144-x^{2}=576$$

Subtract 576 from both sides.

$$72x-144-x^{2}-576=0$$

Subtract 576 from -144 to get -720.

$$72x-720-x^{2}=0$$

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

$$-x^{2}+72x-720=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-720. To find a and b, set up a system to be solved.

$$a+b=72 ab=-\left(-720\right)=720$$

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 720.

$$1,720 2,360 3,240 4,180 5,144 6,120 8,90 9,80 10,72 12,60 15,48 16,45 18,40 20,36 24,30$$

Calculate the sum for each pair.

$$1+720=721 2+360=362 3+240=243 4+180=184 5+144=149 6+120=126 8+90=98 9+80=89 10+72=82 12+60=72 15+48=63 16+45=61 18+40=58 20+36=56 24+30=54$$

The solution is the pair that gives sum 72.

$$a=60 b=12$$

Rewrite -x^{2}+72x-720 as \left(-x^{2}+60x\right)+\left(12x-720\right).

$$\left(-x^{2}+60x\right)+\left(12x-720\right)$$

Factor out -x in the first and 12 in the second group.

$$-x\left(x-60\right)+12\left(x-60\right)$$

Factor out common term x-60 by using distributive property.

$$\left(x-60\right)\left(-x+12\right)$$

To find equation solutions, solve x-60=0 and -x+12=0.

$$x=60 x=12$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$x^{2}-\left(x^{2}-24x+144\right)=\left(x-24\right)^{2}$$

To find the opposite of x^{2}-24x+144, find the opposite of each term.

$$x^{2}-x^{2}+24x-144=\left(x-24\right)^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$24x-144=\left(x-24\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-24\right)^{2}.

$$24x-144=x^{2}-48x+576$$

Subtract x^{2} from both sides.

$$24x-144-x^{2}=-48x+576$$

Add 48x to both sides.

$$24x-144-x^{2}+48x=576$$

Combine 24x and 48x to get 72x.

$$72x-144-x^{2}=576$$

Subtract 576 from both sides.

$$72x-144-x^{2}-576=0$$

Subtract 576 from -144 to get -720.

$$72x-720-x^{2}=0$$

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.

$$-x^{2}+72x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 72 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-1\right)\left(-720\right)}}{2\left(-1\right)}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\left(-1\right)\left(-720\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-72±\sqrt{5184+4\left(-720\right)}}{2\left(-1\right)}$$

Multiply 4 times -720.

$$x=\frac{-72±\sqrt{5184-2880}}{2\left(-1\right)}$$

Add 5184 to -2880.

$$x=\frac{-72±\sqrt{2304}}{2\left(-1\right)}$$

Take the square root of 2304.

$$x=\frac{-72±48}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-72±48}{-2}$$

Now solve the equation x=\frac{-72±48}{-2} when ± is plus. Add -72 to 48.

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

Divide -24 by -2.

$$x=12$$

Now solve the equation x=\frac{-72±48}{-2} when ± is minus. Subtract 48 from -72.

$$x=-\frac{120}{-2}$$

Divide -120 by -2.

$$x=60$$

The equation is now solved.

$$x=12 x=60$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$x^{2}-\left(x^{2}-24x+144\right)=\left(x-24\right)^{2}$$

To find the opposite of x^{2}-24x+144, find the opposite of each term.

$$x^{2}-x^{2}+24x-144=\left(x-24\right)^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$24x-144=\left(x-24\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-24\right)^{2}.

$$24x-144=x^{2}-48x+576$$

Subtract x^{2} from both sides.

$$24x-144-x^{2}=-48x+576$$

Add 48x to both sides.

$$24x-144-x^{2}+48x=576$$

Combine 24x and 48x to get 72x.

$$72x-144-x^{2}=576$$

Add 144 to both sides.

$$72x-x^{2}=576+144$$

Add 576 and 144 to get 720.

$$72x-x^{2}=720$$

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.

$$-x^{2}+72x=720$$

Divide both sides by -1.

$$\frac{-x^{2}+72x}{-1}=\frac{720}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{72}{-1}x=\frac{720}{-1}$$

Divide 72 by -1.

$$x^{2}-72x=\frac{720}{-1}$$

Divide 720 by -1.

$$x^{2}-72x=-720$$

Divide -72, the coefficient of the x term, by 2 to get -36. Then add the square of -36 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-72x+\left(-36\right)^{2}=-720+\left(-36\right)^{2}$$

Square -36.

$$x^{2}-72x+1296=-720+1296$$

Add -720 to 1296.

$$x^{2}-72x+1296=576$$

Factor x^{2}-72x+1296. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-36\right)^{2}=576$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-36\right)^{2}}=\sqrt{576}$$

Simplify.

$$x-36=24 x-36=-24$$

Add 36 to both sides of the equation.

$$x=60 x=12$$