Solve 5b^2-14b-65

$$5 { b }^{ 2 } -14b-65$$

Factor

$$5\left(b-\frac{7-\sqrt{374}}{5}\right)\left(b-\frac{\sqrt{374}+7}{5}\right)$$

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.

$$5b^{2}-14b-65=0$$

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.

$$b=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\left(-65\right)}}{2\times 5}$$

Square -14.

$$b=\frac{-\left(-14\right)±\sqrt{196-4\times 5\left(-65\right)}}{2\times 5}$$

Multiply -4 times 5.

$$b=\frac{-\left(-14\right)±\sqrt{196-20\left(-65\right)}}{2\times 5}$$

Multiply -20 times -65.

$$b=\frac{-\left(-14\right)±\sqrt{196+1300}}{2\times 5}$$

Add 196 to 1300.

$$b=\frac{-\left(-14\right)±\sqrt{1496}}{2\times 5}$$

Take the square root of 1496.

$$b=\frac{-\left(-14\right)±2\sqrt{374}}{2\times 5}$$

The opposite of -14 is 14.

$$b=\frac{14±2\sqrt{374}}{2\times 5}$$

Multiply 2 times 5.

$$b=\frac{14±2\sqrt{374}}{10}$$

Now solve the equation b=\frac{14±2\sqrt{374}}{10} when ± is plus. Add 14 to 2\sqrt{374}.

$$b=\frac{2\sqrt{374}+14}{10}$$

Divide 14+2\sqrt{374} by 10.

$$b=\frac{\sqrt{374}+7}{5}$$

Now solve the equation b=\frac{14±2\sqrt{374}}{10} when ± is minus. Subtract 2\sqrt{374} from 14.

$$b=\frac{14-2\sqrt{374}}{10}$$

Divide 14-2\sqrt{374} by 10.

$$b=\frac{7-\sqrt{374}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{374}}{5} for x_{1} and \frac{7-\sqrt{374}}{5} for x_{2}.

$$5b^{2}-14b-65=5\left(b-\frac{\sqrt{374}+7}{5}\right)\left(b-\frac{7-\sqrt{374}}{5}\right)$$

Evaluate

$$5b^{2}-14b-65$$