Solve z^2-3-1z

$$z ^ { 2 } - 3 - 1 z$$

Factor

$$\left(z-\frac{1-\sqrt{13}}{2}\right)\left(z-\frac{\sqrt{13}+1}{2}\right)$$

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.

$$z^{2}-z-3=0$$

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.

$$z=\frac{-\left(-1\right)±\sqrt{1-4\left(-3\right)}}{2}$$

Multiply -4 times -3.

$$z=\frac{-\left(-1\right)±\sqrt{1+12}}{2}$$

Add 1 to 12.

$$z=\frac{-\left(-1\right)±\sqrt{13}}{2}$$

The opposite of -1 is 1.

$$z=\frac{1±\sqrt{13}}{2}$$

Now solve the equation z=\frac{1±\sqrt{13}}{2} when ± is plus. Add 1 to \sqrt{13}.

$$z=\frac{\sqrt{13}+1}{2}$$

Now solve the equation z=\frac{1±\sqrt{13}}{2} when ± is minus. Subtract \sqrt{13} from 1.

$$z=\frac{1-\sqrt{13}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{13}}{2} for x_{1} and \frac{1-\sqrt{13}}{2} for x_{2}.

$$z^{2}-z-3=\left(z-\frac{\sqrt{13}+1}{2}\right)\left(z-\frac{1-\sqrt{13}}{2}\right)$$

Evaluate

$$z^{2}-z-3$$