Solve 324+3x^2-72x=0

$$324+3 { x }^{ 2 } -72x=0$$

Solve for x

$$x=6$$

$$x=18$$

Steps Using Factoring By Grouping

Divide both sides by 3.

$$108+x^{2}-24x=0$$

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

$$x^{2}-24x+108=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+108. To find a and b, set up a system to be solved.

$$a+b=-24 ab=1\times 108=108$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 108.

$$-1,-108 -2,-54 -3,-36 -4,-27 -6,-18 -9,-12$$

Calculate the sum for each pair.

$$-1-108=-109 -2-54=-56 -3-36=-39 -4-27=-31 -6-18=-24 -9-12=-21$$

The solution is the pair that gives sum -24.

$$a=-18 b=-6$$

Rewrite x^{2}-24x+108 as \left(x^{2}-18x\right)+\left(-6x+108\right).

$$\left(x^{2}-18x\right)+\left(-6x+108\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-18\right)-6\left(x-18\right)$$

Factor out common term x-18 by using distributive property.

$$\left(x-18\right)\left(x-6\right)$$

To find equation solutions, solve x-18=0 and x-6=0.

$$x=18 x=6$$

Steps Using the Quadratic Formula

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.

$$3x^{2}-72x+324=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -72 for b, and 324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 3\times 324}}{2\times 3}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 3\times 324}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-72\right)±\sqrt{5184-12\times 324}}{2\times 3}$$

Multiply -12 times 324.

$$x=\frac{-\left(-72\right)±\sqrt{5184-3888}}{2\times 3}$$

Add 5184 to -3888.

$$x=\frac{-\left(-72\right)±\sqrt{1296}}{2\times 3}$$

Take the square root of 1296.

$$x=\frac{-\left(-72\right)±36}{2\times 3}$$

The opposite of -72 is 72.

$$x=\frac{72±36}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{72±36}{6}$$

Now solve the equation x=\frac{72±36}{6} when ± is plus. Add 72 to 36.

$$x=\frac{108}{6}$$

Divide 108 by 6.

$$x=18$$

Now solve the equation x=\frac{72±36}{6} when ± is minus. Subtract 36 from 72.

$$x=\frac{36}{6}$$

Divide 36 by 6.

$$x=6$$

The equation is now solved.

$$x=18 x=6$$

Steps for Completing the Square

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.

$$3x^{2}-72x+324=0$$

Subtract 324 from both sides of the equation.

$$3x^{2}-72x+324-324=-324$$

Subtracting 324 from itself leaves 0.

$$3x^{2}-72x=-324$$

Divide both sides by 3.

$$\frac{3x^{2}-72x}{3}=-\frac{324}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{72}{3}\right)x=-\frac{324}{3}$$

Divide -72 by 3.

$$x^{2}-24x=-\frac{324}{3}$$

Divide -324 by 3.

$$x^{2}-24x=-108$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=-108+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-108+144$$

Add -108 to 144.

$$x^{2}-24x+144=36$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-12=6 x-12=-6$$

Add 12 to both sides of the equation.

$$x=18 x=6$$