Solve 24+4left(x-2right)-xleft(x-2right)=0

$$24+4 \left( x-2 \right) -x \left( x-2 \right) = 0$$

Solve for x

$$x=-2$$

$$x=8$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 4 by x-2.

$$24+4x-8-x\left(x-2\right)=0$$

Subtract 8 from 24 to get 16.

$$16+4x-x\left(x-2\right)=0$$

Use the distributive property to multiply x by x-2.

$$16+4x-\left(x^{2}-2x\right)=0$$

To find the opposite of x^{2}-2x, find the opposite of each term.

$$16+4x-x^{2}-\left(-2x\right)=0$$

The opposite of -2x is 2x.

$$16+4x-x^{2}+2x=0$$

Combine 4x and 2x to get 6x.

$$16+6x-x^{2}=0$$

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

$$-x^{2}+6x+16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=6 ab=-16=-16$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 6.

$$a=8 b=-2$$

Rewrite -x^{2}+6x+16 as \left(-x^{2}+8x\right)+\left(-2x+16\right).

$$\left(-x^{2}+8x\right)+\left(-2x+16\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-8\right)-2\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(-x-2\right)$$

To find equation solutions, solve x-8=0 and -x-2=0.

$$x=8 x=-2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by x-2.

$$24+4x-8-x\left(x-2\right)=0$$

Subtract 8 from 24 to get 16.

$$16+4x-x\left(x-2\right)=0$$

Use the distributive property to multiply x by x-2.

$$16+4x-\left(x^{2}-2x\right)=0$$

To find the opposite of x^{2}-2x, find the opposite of each term.

$$16+4x-x^{2}-\left(-2x\right)=0$$

The opposite of -2x is 2x.

$$16+4x-x^{2}+2x=0$$

Combine 4x and 2x to get 6x.

$$16+6x-x^{2}=0$$

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.

$$-x^{2}+6x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\times 16}}{2\left(-1\right)}$$

Multiply 4 times 16.

$$x=\frac{-6±\sqrt{36+64}}{2\left(-1\right)}$$

Add 36 to 64.

$$x=\frac{-6±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-6±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±10}{-2}$$

Now solve the equation x=\frac{-6±10}{-2} when ± is plus. Add -6 to 10.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-6±10}{-2} when ± is minus. Subtract 10 from -6.

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

Divide -16 by -2.

$$x=8$$

The equation is now solved.

$$x=-2 x=8$$

Steps for Completing the Square

Use the distributive property to multiply 4 by x-2.

$$24+4x-8-x\left(x-2\right)=0$$

Subtract 8 from 24 to get 16.

$$16+4x-x\left(x-2\right)=0$$

Use the distributive property to multiply x by x-2.

$$16+4x-\left(x^{2}-2x\right)=0$$

To find the opposite of x^{2}-2x, find the opposite of each term.

$$16+4x-x^{2}-\left(-2x\right)=0$$

The opposite of -2x is 2x.

$$16+4x-x^{2}+2x=0$$

Combine 4x and 2x to get 6x.

$$16+6x-x^{2}=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$6x-x^{2}=-16$$

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.

$$-x^{2}+6x=-16$$

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=-\frac{16}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=-\frac{16}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=-\frac{16}{-1}$$

Divide -16 by -1.

$$x^{2}-6x=16$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=16+9$$

Add 16 to 9.

$$x^{2}-6x+9=25$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-3=5 x-3=-5$$

Add 3 to both sides of the equation.

$$x=8 x=-2$$