Solve x^2+96=20x

$${ x }^{ 2 } +96=20x$$

Solve for x

$$x=8$$

$$x=12$$

Steps Using Factoring

Subtract 20x from both sides.

$$x^{2}+96-20x=0$$

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

$$x^{2}-20x+96=0$$

To solve the equation, factor x^{2}-20x+96 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.

$$a+b=-20 ab=96$$

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 96.

$$-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12$$

Calculate the sum for each pair.

$$-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20$$

The solution is the pair that gives sum -20.

$$a=-12 b=-8$$

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

$$\left(x-12\right)\left(x-8\right)$$

To find equation solutions, solve x-12=0 and x-8=0.

$$x=12 x=8$$

Steps Using Factoring By Grouping

Subtract 20x from both sides.

$$x^{2}+96-20x=0$$

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

$$x^{2}-20x+96=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+96. To find a and b, set up a system to be solved.

$$a+b=-20 ab=1\times 96=96$$

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 96.

$$-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12$$

Calculate the sum for each pair.

$$-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20$$

The solution is the pair that gives sum -20.

$$a=-12 b=-8$$

Rewrite x^{2}-20x+96 as \left(x^{2}-12x\right)+\left(-8x+96\right).

$$\left(x^{2}-12x\right)+\left(-8x+96\right)$$

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

$$x\left(x-12\right)-8\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x-8\right)$$

To find equation solutions, solve x-12=0 and x-8=0.

$$x=12 x=8$$

Steps Using the Quadratic Formula

Subtract 20x from both sides.

$$x^{2}+96-20x=0$$

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.

$$x^{2}-20x+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 96}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 96}}{2}$$

Multiply -4 times 96.

$$x=\frac{-\left(-20\right)±\sqrt{400-384}}{2}$$

Add 400 to -384.

$$x=\frac{-\left(-20\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-20\right)±4}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±4}{2}$$

Now solve the equation x=\frac{20±4}{2} when ± is plus. Add 20 to 4.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{20±4}{2} when ± is minus. Subtract 4 from 20.

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

Divide 16 by 2.

$$x=8$$

The equation is now solved.

$$x=12 x=8$$

Steps for Completing the Square

Subtract 20x from both sides.

$$x^{2}+96-20x=0$$

Subtract 96 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-20x=-96$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-96+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-96+100$$

Add -96 to 100.

$$x^{2}-20x+100=4$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-10=2 x-10=-2$$

Add 10 to both sides of the equation.

$$x=12 x=8$$