Solve 4a^2-11a-5

$$4 { a }^{ 2 } -11a-5$$

Factor

$$4\left(a-\frac{11-\sqrt{201}}{8}\right)\left(a-\frac{\sqrt{201}+11}{8}\right)$$

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.

$$4a^{2}-11a-5=0$$

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.

$$a=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\left(-5\right)}}{2\times 4}$$

Square -11.

$$a=\frac{-\left(-11\right)±\sqrt{121-4\times 4\left(-5\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-11\right)±\sqrt{121-16\left(-5\right)}}{2\times 4}$$

Multiply -16 times -5.

$$a=\frac{-\left(-11\right)±\sqrt{121+80}}{2\times 4}$$

Add 121 to 80.

$$a=\frac{-\left(-11\right)±\sqrt{201}}{2\times 4}$$

The opposite of -11 is 11.

$$a=\frac{11±\sqrt{201}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{11±\sqrt{201}}{8}$$

Now solve the equation a=\frac{11±\sqrt{201}}{8} when ± is plus. Add 11 to \sqrt{201}.

$$a=\frac{\sqrt{201}+11}{8}$$

Now solve the equation a=\frac{11±\sqrt{201}}{8} when ± is minus. Subtract \sqrt{201} from 11.

$$a=\frac{11-\sqrt{201}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{201}}{8} for x_{1} and \frac{11-\sqrt{201}}{8} for x_{2}.

$$4a^{2}-11a-5=4\left(a-\frac{\sqrt{201}+11}{8}\right)\left(a-\frac{11-\sqrt{201}}{8}\right)$$

Evaluate

$$4a^{2}-11a-5$$