Solve 23x=120+x^2

$$23x=120+ { x }^{ 2 }$$

Solve for x

$$x=8$$

$$x=15$$

Steps Using Factoring By Grouping

Subtract 120 from both sides.

$$23x-120=x^{2}$$

Subtract x^{2} from both sides.

$$23x-120-x^{2}=0$$

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

$$-x^{2}+23x-120=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-120. To find a and b, set up a system to be solved.

$$a+b=23 ab=-\left(-120\right)=120$$

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 120.

$$1,120 2,60 3,40 4,30 5,24 6,20 8,15 10,12$$

Calculate the sum for each pair.

$$1+120=121 2+60=62 3+40=43 4+30=34 5+24=29 6+20=26 8+15=23 10+12=22$$

The solution is the pair that gives sum 23.

$$a=15 b=8$$

Rewrite -x^{2}+23x-120 as \left(-x^{2}+15x\right)+\left(8x-120\right).

$$\left(-x^{2}+15x\right)+\left(8x-120\right)$$

Factor out -x in the first and 8 in the second group.

$$-x\left(x-15\right)+8\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(-x+8\right)$$

To find equation solutions, solve x-15=0 and -x+8=0.

$$x=15 x=8$$

Steps Using the Quadratic Formula

Subtract 120 from both sides.

$$23x-120=x^{2}$$

Subtract x^{2} from both sides.

$$23x-120-x^{2}=0$$

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.

$$-x^{2}+23x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 23 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23±\sqrt{23^{2}-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-23±\sqrt{529+4\left(-120\right)}}{2\left(-1\right)}$$

Multiply 4 times -120.

$$x=\frac{-23±\sqrt{529-480}}{2\left(-1\right)}$$

Add 529 to -480.

$$x=\frac{-23±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-23±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-23±7}{-2}$$

Now solve the equation x=\frac{-23±7}{-2} when ± is plus. Add -23 to 7.

$$x=-\frac{16}{-2}$$

Divide -16 by -2.

$$x=8$$

Now solve the equation x=\frac{-23±7}{-2} when ± is minus. Subtract 7 from -23.

$$x=-\frac{30}{-2}$$

Divide -30 by -2.

$$x=15$$

The equation is now solved.

$$x=8 x=15$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$23x-x^{2}=120$$

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.

$$-x^{2}+23x=120$$

Divide both sides by -1.

$$\frac{-x^{2}+23x}{-1}=\frac{120}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{23}{-1}x=\frac{120}{-1}$$

Divide 23 by -1.

$$x^{2}-23x=\frac{120}{-1}$$

Divide 120 by -1.

$$x^{2}-23x=-120$$

Divide -23, the coefficient of the x term, by 2 to get -\frac{23}{2}. Then add the square of -\frac{23}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-23x+\left(-\frac{23}{2}\right)^{2}=-120+\left(-\frac{23}{2}\right)^{2}$$

Square -\frac{23}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-23x+\frac{529}{4}=-120+\frac{529}{4}$$

Add -120 to \frac{529}{4}.

$$x^{2}-23x+\frac{529}{4}=\frac{49}{4}$$

Factor x^{2}-23x+\frac{529}{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}.

$$\left(x-\frac{23}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{23}{2}=\frac{7}{2} x-\frac{23}{2}=-\frac{7}{2}$$

Add \frac{23}{2} to both sides of the equation.

$$x=15 x=8$$