Solve =a^2-7a-4

$$= { a }^{ 2 } -7a-4$$

Factor

$$\left(a-\frac{7-\sqrt{65}}{2}\right)\left(a-\frac{\sqrt{65}+7}{2}\right)$$

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.

$$a^{2}-7a-4=0$$

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.

$$a=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-4\right)}}{2}$$

Square -7.

$$a=\frac{-\left(-7\right)±\sqrt{49-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$a=\frac{-\left(-7\right)±\sqrt{49+16}}{2}$$

Add 49 to 16.

$$a=\frac{-\left(-7\right)±\sqrt{65}}{2}$$

The opposite of -7 is 7.

$$a=\frac{7±\sqrt{65}}{2}$$

Now solve the equation a=\frac{7±\sqrt{65}}{2} when ± is plus. Add 7 to \sqrt{65}.

$$a=\frac{\sqrt{65}+7}{2}$$

Now solve the equation a=\frac{7±\sqrt{65}}{2} when ± is minus. Subtract \sqrt{65} from 7.

$$a=\frac{7-\sqrt{65}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{65}}{2} for x_{1} and \frac{7-\sqrt{65}}{2} for x_{2}.

$$a^{2}-7a-4=\left(a-\frac{\sqrt{65}+7}{2}\right)\left(a-\frac{7-\sqrt{65}}{2}\right)$$

Evaluate

$$a^{2}-7a-4$$