Solve 2x^2-6x+4

$$2 { x }^{ 2 } -6x+4$$

Factor

$$2\left(x-2\right)\left(x-1\right)$$

Solution Steps

Factor out 2.

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

Consider x^{2}-3x+2. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite x^{2}-3x+2 as \left(x^{2}-2x\right)+\left(-x+2\right).

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

Factor out x in the first and -1 in the second group.

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

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$2\left(x-2\right)\left(x-1\right)$$

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.

$$2x^{2}-6x+4=0$$

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.

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

Square -6.

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

Multiply -4 times 2.

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

Multiply -8 times 4.

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

Add 36 to -32.

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

Take the square root of 4.

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

The opposite of -6 is 6.

$$x=\frac{6±2}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{6±2}{4}$$

Now solve the equation x=\frac{6±2}{4} when ± is plus. Add 6 to 2.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{6±2}{4} when ± is minus. Subtract 2 from 6.

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

Divide 4 by 4.

$$x=1$$

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

$$2x^{2}-6x+4=2\left(x-2\right)\left(x-1\right)$$

Evaluate

$$2\left(x-2\right)\left(x-1\right)$$