Solve 15x^2-141x+90

$$15 { x }^{ 2 } -141x+90$$

Factor

$$15\left(x-\frac{47-\sqrt{1609}}{10}\right)\left(x-\frac{\sqrt{1609}+47}{10}\right)$$

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.

$$15x^{2}-141x+90=0$$

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.

$$x=\frac{-\left(-141\right)±\sqrt{\left(-141\right)^{2}-4\times 15\times 90}}{2\times 15}$$

Square -141.

$$x=\frac{-\left(-141\right)±\sqrt{19881-4\times 15\times 90}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-141\right)±\sqrt{19881-60\times 90}}{2\times 15}$$

Multiply -60 times 90.

$$x=\frac{-\left(-141\right)±\sqrt{19881-5400}}{2\times 15}$$

Add 19881 to -5400.

$$x=\frac{-\left(-141\right)±\sqrt{14481}}{2\times 15}$$

Take the square root of 14481.

$$x=\frac{-\left(-141\right)±3\sqrt{1609}}{2\times 15}$$

The opposite of -141 is 141.

$$x=\frac{141±3\sqrt{1609}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{141±3\sqrt{1609}}{30}$$

Now solve the equation x=\frac{141±3\sqrt{1609}}{30} when ± is plus. Add 141 to 3\sqrt{1609}.

$$x=\frac{3\sqrt{1609}+141}{30}$$

Divide 141+3\sqrt{1609} by 30.

$$x=\frac{\sqrt{1609}+47}{10}$$

Now solve the equation x=\frac{141±3\sqrt{1609}}{30} when ± is minus. Subtract 3\sqrt{1609} from 141.

$$x=\frac{141-3\sqrt{1609}}{30}$$

Divide 141-3\sqrt{1609} by 30.

$$x=\frac{47-\sqrt{1609}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{47+\sqrt{1609}}{10} for x_{1} and \frac{47-\sqrt{1609}}{10} for x_{2}.

$$15x^{2}-141x+90=15\left(x-\frac{\sqrt{1609}+47}{10}\right)\left(x-\frac{47-\sqrt{1609}}{10}\right)$$

Evaluate

$$15x^{2}-141x+90$$