Solve x^2-60x+864=0

$${ x }^{ 2 } -60x+864 = 0$$

Solve for x

$$x=24$$

$$x=36$$

Steps Using Factoring

To solve the equation, factor x^{2}-60x+864 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-60 ab=864$$

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 864.

$$-1,-864 -2,-432 -3,-288 -4,-216 -6,-144 -8,-108 -9,-96 -12,-72 -16,-54 -18,-48 -24,-36 -27,-32$$

Calculate the sum for each pair.

$$-1-864=-865 -2-432=-434 -3-288=-291 -4-216=-220 -6-144=-150 -8-108=-116 -9-96=-105 -12-72=-84 -16-54=-70 -18-48=-66 -24-36=-60 -27-32=-59$$

The solution is the pair that gives sum -60.

$$a=-36 b=-24$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-36\right)\left(x-24\right)$$

To find equation solutions, solve x-36=0 and x-24=0.

$$x=36 x=24$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+864. To find a and b, set up a system to be solved.

$$a+b=-60 ab=1\times 864=864$$

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 864.

$$-1,-864 -2,-432 -3,-288 -4,-216 -6,-144 -8,-108 -9,-96 -12,-72 -16,-54 -18,-48 -24,-36 -27,-32$$

Calculate the sum for each pair.

$$-1-864=-865 -2-432=-434 -3-288=-291 -4-216=-220 -6-144=-150 -8-108=-116 -9-96=-105 -12-72=-84 -16-54=-70 -18-48=-66 -24-36=-60 -27-32=-59$$

The solution is the pair that gives sum -60.

$$a=-36 b=-24$$

Rewrite x^{2}-60x+864 as \left(x^{2}-36x\right)+\left(-24x+864\right).

$$\left(x^{2}-36x\right)+\left(-24x+864\right)$$

Factor out x in the first and -24 in the second group.

$$x\left(x-36\right)-24\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(x-24\right)$$

To find equation solutions, solve x-36=0 and x-24=0.

$$x=36 x=24$$

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.

$$x^{2}-60x+864=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -60 for b, and 864 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 864}}{2}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 864}}{2}$$

Multiply -4 times 864.

$$x=\frac{-\left(-60\right)±\sqrt{3600-3456}}{2}$$

Add 3600 to -3456.

$$x=\frac{-\left(-60\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-\left(-60\right)±12}{2}$$

The opposite of -60 is 60.

$$x=\frac{60±12}{2}$$

Now solve the equation x=\frac{60±12}{2} when ± is plus. Add 60 to 12.

$$x=\frac{72}{2}$$

Divide 72 by 2.

$$x=36$$

Now solve the equation x=\frac{60±12}{2} when ± is minus. Subtract 12 from 60.

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

Divide 48 by 2.

$$x=24$$

The equation is now solved.

$$x=36 x=24$$

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.

$$x^{2}-60x+864=0$$

Subtract 864 from both sides of the equation.

$$x^{2}-60x+864-864=-864$$

Subtracting 864 from itself leaves 0.

$$x^{2}-60x=-864$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-864+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-864+900$$

Add -864 to 900.

$$x^{2}-60x+900=36$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-30=6 x-30=-6$$

Add 30 to both sides of the equation.

$$x=36 x=24$$