Solve x^2-x-5

$${ x }^{ 2 } -x-5$$

Factor

$$\left(x-\frac{1-\sqrt{21}}{2}\right)\left(x-\frac{\sqrt{21}+1}{2}\right)$$

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.

$$x^{2}-x-5=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$x=\frac{-\left(-1\right)±\sqrt{1+20}}{2}$$

Add 1 to 20.

$$x=\frac{-\left(-1\right)±\sqrt{21}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{21}}{2}$$

Now solve the equation x=\frac{1±\sqrt{21}}{2} when ± is plus. Add 1 to \sqrt{21}.

$$x=\frac{\sqrt{21}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{21}}{2} when ± is minus. Subtract \sqrt{21} from 1.

$$x=\frac{1-\sqrt{21}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{21}}{2} for x_{1} and \frac{1-\sqrt{21}}{2} for x_{2}.

$$x^{2}-x-5=\left(x-\frac{\sqrt{21}+1}{2}\right)\left(x-\frac{1-\sqrt{21}}{2}\right)$$

Evaluate

$$x^{2}-x-5$$