Solve x+12=x^2-2*x*18+18^2

$$x+12= { x }^{ 2 } -2 \times x \times 18+ { 18 }^{ 2 }$$

Solve for x

$$x=13$$

$$x=24$$

Steps Using Factoring By Grouping

Multiply 2 and 18 to get 36.

$$x+12=x^{2}-36x+18^{2}$$

Calculate 18 to the power of 2 and get 324.

$$x+12=x^{2}-36x+324$$

Subtract x^{2} from both sides.

$$x+12-x^{2}=-36x+324$$

Add 36x to both sides.

$$x+12-x^{2}+36x=324$$

Combine x and 36x to get 37x.

$$37x+12-x^{2}=324$$

Subtract 324 from both sides.

$$37x+12-x^{2}-324=0$$

Subtract 324 from 12 to get -312.

$$37x-312-x^{2}=0$$

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

$$-x^{2}+37x-312=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-312. To find a and b, set up a system to be solved.

$$a+b=37 ab=-\left(-312\right)=312$$

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 312.

$$1,312 2,156 3,104 4,78 6,52 8,39 12,26 13,24$$

Calculate the sum for each pair.

$$1+312=313 2+156=158 3+104=107 4+78=82 6+52=58 8+39=47 12+26=38 13+24=37$$

The solution is the pair that gives sum 37.

$$a=24 b=13$$

Rewrite -x^{2}+37x-312 as \left(-x^{2}+24x\right)+\left(13x-312\right).

$$\left(-x^{2}+24x\right)+\left(13x-312\right)$$

Factor out -x in the first and 13 in the second group.

$$-x\left(x-24\right)+13\left(x-24\right)$$

Factor out common term x-24 by using distributive property.

$$\left(x-24\right)\left(-x+13\right)$$

To find equation solutions, solve x-24=0 and -x+13=0.

$$x=24 x=13$$

Steps Using the Quadratic Formula

Multiply 2 and 18 to get 36.

$$x+12=x^{2}-36x+18^{2}$$

Calculate 18 to the power of 2 and get 324.

$$x+12=x^{2}-36x+324$$

Subtract x^{2} from both sides.

$$x+12-x^{2}=-36x+324$$

Add 36x to both sides.

$$x+12-x^{2}+36x=324$$

Combine x and 36x to get 37x.

$$37x+12-x^{2}=324$$

Subtract 324 from both sides.

$$37x+12-x^{2}-324=0$$

Subtract 324 from 12 to get -312.

$$37x-312-x^{2}=0$$

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.

$$-x^{2}+37x-312=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 37 for b, and -312 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-37±\sqrt{37^{2}-4\left(-1\right)\left(-312\right)}}{2\left(-1\right)}$$

Square 37.

$$x=\frac{-37±\sqrt{1369-4\left(-1\right)\left(-312\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-37±\sqrt{1369+4\left(-312\right)}}{2\left(-1\right)}$$

Multiply 4 times -312.

$$x=\frac{-37±\sqrt{1369-1248}}{2\left(-1\right)}$$

Add 1369 to -1248.

$$x=\frac{-37±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-37±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-37±11}{-2}$$

Now solve the equation x=\frac{-37±11}{-2} when ± is plus. Add -37 to 11.

$$x=-\frac{26}{-2}$$

Divide -26 by -2.

$$x=13$$

Now solve the equation x=\frac{-37±11}{-2} when ± is minus. Subtract 11 from -37.

$$x=-\frac{48}{-2}$$

Divide -48 by -2.

$$x=24$$

The equation is now solved.

$$x=13 x=24$$

Steps for Completing the Square

Multiply 2 and 18 to get 36.

$$x+12=x^{2}-36x+18^{2}$$

Calculate 18 to the power of 2 and get 324.

$$x+12=x^{2}-36x+324$$

Subtract x^{2} from both sides.

$$x+12-x^{2}=-36x+324$$

Add 36x to both sides.

$$x+12-x^{2}+36x=324$$

Combine x and 36x to get 37x.

$$37x+12-x^{2}=324$$

Subtract 12 from both sides.

$$37x-x^{2}=324-12$$

Subtract 12 from 324 to get 312.

$$37x-x^{2}=312$$

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.

$$-x^{2}+37x=312$$

Divide both sides by -1.

$$\frac{-x^{2}+37x}{-1}=\frac{312}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{37}{-1}x=\frac{312}{-1}$$

Divide 37 by -1.

$$x^{2}-37x=\frac{312}{-1}$$

Divide 312 by -1.

$$x^{2}-37x=-312$$

Divide -37, the coefficient of the x term, by 2 to get -\frac{37}{2}. Then add the square of -\frac{37}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-37x+\left(-\frac{37}{2}\right)^{2}=-312+\left(-\frac{37}{2}\right)^{2}$$

Square -\frac{37}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-37x+\frac{1369}{4}=-312+\frac{1369}{4}$$

Add -312 to \frac{1369}{4}.

$$x^{2}-37x+\frac{1369}{4}=\frac{121}{4}$$

Factor x^{2}-37x+\frac{1369}{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}.

$$\left(x-\frac{37}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{37}{2}=\frac{11}{2} x-\frac{37}{2}=-\frac{11}{2}$$

Add \frac{37}{2} to both sides of the equation.

$$x=24 x=13$$