Solve 2x^2-48x+9*24=0

$$2 x ^ { 2 } - 48 x + 9 \times 24 = 0$$

Solve for x

$$x=6$$

$$x=18$$

Steps Using Factoring By Grouping

Multiply 9 and 24 to get 216.

$$2x^{2}-48x+216=0$$

Divide both sides by 2.

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

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

Multiply 9 and 24 to get 216.

$$2x^{2}-48x+216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -48 for b, and 216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 2\times 216}}{2\times 2}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 2\times 216}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-48\right)±\sqrt{2304-8\times 216}}{2\times 2}$$

Multiply -8 times 216.

$$x=\frac{-\left(-48\right)±\sqrt{2304-1728}}{2\times 2}$$

Add 2304 to -1728.

$$x=\frac{-\left(-48\right)±\sqrt{576}}{2\times 2}$$

Take the square root of 576.

$$x=\frac{-\left(-48\right)±24}{2\times 2}$$

The opposite of -48 is 48.

$$x=\frac{48±24}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{48±24}{4}$$

Now solve the equation x=\frac{48±24}{4} when ± is plus. Add 48 to 24.

$$x=\frac{72}{4}$$

Divide 72 by 4.

$$x=18$$

Now solve the equation x=\frac{48±24}{4} when ± is minus. Subtract 24 from 48.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

The equation is now solved.

$$x=18 x=6$$

Steps for Completing the Square

Multiply 9 and 24 to get 216.

$$2x^{2}-48x+216=0$$

Subtract 216 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}-48x=-216$$

Divide both sides by 2.

$$\frac{2x^{2}-48x}{2}=-\frac{216}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{48}{2}\right)x=-\frac{216}{2}$$

Divide -48 by 2.

$$x^{2}-24x=-\frac{216}{2}$$

Divide -216 by 2.

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

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.

$$x=18 x=6$$