Solve 15x^2-3x-14

$$15 { x }^{ 2 } -3x-14$$

Factor

$$15\left(x-\left(-\frac{\sqrt{849}}{30}+\frac{1}{10}\right)\right)\left(x-\left(\frac{\sqrt{849}}{30}+\frac{1}{10}\right)\right)$$

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.

$$15x^{2}-3x-14=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 15\left(-14\right)}}{2\times 15}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 15\left(-14\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-3\right)±\sqrt{9-60\left(-14\right)}}{2\times 15}$$

Multiply -60 times -14.

$$x=\frac{-\left(-3\right)±\sqrt{9+840}}{2\times 15}$$

Add 9 to 840.

$$x=\frac{-\left(-3\right)±\sqrt{849}}{2\times 15}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{849}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{3±\sqrt{849}}{30}$$

Now solve the equation x=\frac{3±\sqrt{849}}{30} when ± is plus. Add 3 to \sqrt{849}.

$$x=\frac{\sqrt{849}+3}{30}$$

Divide 3+\sqrt{849} by 30.

$$x=\frac{\sqrt{849}}{30}+\frac{1}{10}$$

Now solve the equation x=\frac{3±\sqrt{849}}{30} when ± is minus. Subtract \sqrt{849} from 3.

$$x=\frac{3-\sqrt{849}}{30}$$

Divide 3-\sqrt{849} by 30.

$$x=-\frac{\sqrt{849}}{30}+\frac{1}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{10}+\frac{\sqrt{849}}{30} for x_{1} and \frac{1}{10}-\frac{\sqrt{849}}{30} for x_{2}.

$$15x^{2}-3x-14=15\left(x-\left(\frac{\sqrt{849}}{30}+\frac{1}{10}\right)\right)\left(x-\left(-\frac{\sqrt{849}}{30}+\frac{1}{10}\right)\right)$$

Evaluate

$$15x^{2}-3x-14$$