Solve 5x^2-23x+15

$$5 { x }^{ 2 } -23x+15$$

Factor

$$5\left(x-\frac{23-\sqrt{229}}{10}\right)\left(x-\frac{\sqrt{229}+23}{10}\right)$$

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.

$$5x^{2}-23x+15=0$$

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.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 5\times 15}}{2\times 5}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 5\times 15}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-23\right)±\sqrt{529-20\times 15}}{2\times 5}$$

Multiply -20 times 15.

$$x=\frac{-\left(-23\right)±\sqrt{529-300}}{2\times 5}$$

Add 529 to -300.

$$x=\frac{-\left(-23\right)±\sqrt{229}}{2\times 5}$$

The opposite of -23 is 23.

$$x=\frac{23±\sqrt{229}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{23±\sqrt{229}}{10}$$

Now solve the equation x=\frac{23±\sqrt{229}}{10} when ± is plus. Add 23 to \sqrt{229}.

$$x=\frac{\sqrt{229}+23}{10}$$

Now solve the equation x=\frac{23±\sqrt{229}}{10} when ± is minus. Subtract \sqrt{229} from 23.

$$x=\frac{23-\sqrt{229}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{23+\sqrt{229}}{10} for x_{1} and \frac{23-\sqrt{229}}{10} for x_{2}.

$$5x^{2}-23x+15=5\left(x-\frac{\sqrt{229}+23}{10}\right)\left(x-\frac{23-\sqrt{229}}{10}\right)$$

Evaluate

$$5x^{2}-23x+15$$