Solve x^2+-10x+20

$$x ^ { 2 } + - 10 x + 20$$

Factor

$$\left(x-\left(5-\sqrt{5}\right)\right)\left(x-\left(\sqrt{5}+5\right)\right)$$

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.

$$x^{2}-10x+20=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 20}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 20}}{2}$$

Multiply -4 times 20.

$$x=\frac{-\left(-10\right)±\sqrt{100-80}}{2}$$

Add 100 to -80.

$$x=\frac{-\left(-10\right)±\sqrt{20}}{2}$$

Take the square root of 20.

$$x=\frac{-\left(-10\right)±2\sqrt{5}}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{5}}{2}$$

Now solve the equation x=\frac{10±2\sqrt{5}}{2} when ± is plus. Add 10 to 2\sqrt{5}.

$$x=\frac{2\sqrt{5}+10}{2}$$

Divide 10+2\sqrt{5} by 2.

$$x=\sqrt{5}+5$$

Now solve the equation x=\frac{10±2\sqrt{5}}{2} when ± is minus. Subtract 2\sqrt{5} from 10.

$$x=\frac{10-2\sqrt{5}}{2}$$

Divide 10-2\sqrt{5} by 2.

$$x=5-\sqrt{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5+\sqrt{5} for x_{1} and 5-\sqrt{5} for x_{2}.

$$x^{2}-10x+20=\left(x-\left(\sqrt{5}+5\right)\right)\left(x-\left(5-\sqrt{5}\right)\right)$$

Evaluate

$$x^{2}-10x+20$$