Solve x^2+80=24x

$${ x }^{ 2 } +80 = 24x$$

Solve for x

$$x=4$$

$$x=20$$

Steps Using Factoring

Subtract 24x from both sides.

$$x^{2}+80-24x=0$$

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

$$x^{2}-24x+80=0$$

To solve the equation, factor x^{2}-24x+80 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.

$$a+b=-24 ab=80$$

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 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Calculate the sum for each pair.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

The solution is the pair that gives sum -24.

$$a=-20 b=-4$$

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

$$\left(x-20\right)\left(x-4\right)$$

To find equation solutions, solve x-20=0 and x-4=0.

$$x=20 x=4$$

Steps Using Factoring By Grouping

Subtract 24x from both sides.

$$x^{2}+80-24x=0$$

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

$$x^{2}-24x+80=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+80. To find a and b, set up a system to be solved.

$$a+b=-24 ab=1\times 80=80$$

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 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Calculate the sum for each pair.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

The solution is the pair that gives sum -24.

$$a=-20 b=-4$$

Rewrite x^{2}-24x+80 as \left(x^{2}-20x\right)+\left(-4x+80\right).

$$\left(x^{2}-20x\right)+\left(-4x+80\right)$$

Factor out x in the first and -4 in the second group.

$$x\left(x-20\right)-4\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(x-4\right)$$

To find equation solutions, solve x-20=0 and x-4=0.

$$x=20 x=4$$

Steps Using the Quadratic Formula

Subtract 24x from both sides.

$$x^{2}+80-24x=0$$

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.

$$x^{2}-24x+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 80}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 80}}{2}$$

Multiply -4 times 80.

$$x=\frac{-\left(-24\right)±\sqrt{576-320}}{2}$$

Add 576 to -320.

$$x=\frac{-\left(-24\right)±\sqrt{256}}{2}$$

Take the square root of 256.

$$x=\frac{-\left(-24\right)±16}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±16}{2}$$

Now solve the equation x=\frac{24±16}{2} when ± is plus. Add 24 to 16.

$$x=\frac{40}{2}$$

Divide 40 by 2.

$$x=20$$

Now solve the equation x=\frac{24±16}{2} when ± is minus. Subtract 16 from 24.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

The equation is now solved.

$$x=20 x=4$$

Steps for Completing the Square

Subtract 24x from both sides.

$$x^{2}+80-24x=0$$

Subtract 80 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-24x=-80$$

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.

$$x^{2}-24x+\left(-12\right)^{2}=-80+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-80+144$$

Add -80 to 144.

$$x^{2}-24x+144=64$$

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}.

$$\left(x-12\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-12=8 x-12=-8$$

Add 12 to both sides of the equation.

$$x=20 x=4$$