Solve x^2-6x+8-0

$$x ^ { 2 } - 6 x + 8 - 0$$

Evaluate

$$\left(x-4\right)\left(x-2\right)$$

Solution Steps

Multiply -1 and 0 to get 0.

$$x^{2}-6x+8+0$$

Add 8 and 0 to get 8.

$$x^{2}-6x+8$$

Factor

$$\left(x-4\right)\left(x-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\times 8=8$$

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 8.

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

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -6.

$$a=-4 b=-2$$

Rewrite x^{2}-6x+8 as \left(x^{2}-4x\right)+\left(-2x+8\right).

$$\left(x^{2}-4x\right)+\left(-2x+8\right)$$

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

$$x\left(x-4\right)-2\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x-2\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-6x+8=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-\left(-6\right)±\sqrt{36-32}}{2}$$

Add 36 to -32.

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

Take the square root of 4.

$$x=\frac{-\left(-6\right)±2}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2}{2}$$

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

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

Divide 8 by 2.

$$x=4$$

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

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

Divide 4 by 2.

$$x=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and 2 for x_{2}.

$$x^{2}-6x+8=\left(x-4\right)\left(x-2\right)$$