Solve 6x^2+500x-1665

$$6 { x }^{ 2 } +500x-1665$$

Factor

$$6\left(x-\left(-\frac{\sqrt{72490}}{6}-\frac{125}{3}\right)\right)\left(x-\left(\frac{\sqrt{72490}}{6}-\frac{125}{3}\right)\right)$$

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.

$$6x^{2}+500x-1665=0$$

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.

$$x=\frac{-500±\sqrt{500^{2}-4\times 6\left(-1665\right)}}{2\times 6}$$

Square 500.

$$x=\frac{-500±\sqrt{250000-4\times 6\left(-1665\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-500±\sqrt{250000-24\left(-1665\right)}}{2\times 6}$$

Multiply -24 times -1665.

$$x=\frac{-500±\sqrt{250000+39960}}{2\times 6}$$

Add 250000 to 39960.

$$x=\frac{-500±\sqrt{289960}}{2\times 6}$$

Take the square root of 289960.

$$x=\frac{-500±2\sqrt{72490}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-500±2\sqrt{72490}}{12}$$

Now solve the equation x=\frac{-500±2\sqrt{72490}}{12} when ± is plus. Add -500 to 2\sqrt{72490}.

$$x=\frac{2\sqrt{72490}-500}{12}$$

Divide -500+2\sqrt{72490} by 12.

$$x=\frac{\sqrt{72490}}{6}-\frac{125}{3}$$

Now solve the equation x=\frac{-500±2\sqrt{72490}}{12} when ± is minus. Subtract 2\sqrt{72490} from -500.

$$x=\frac{-2\sqrt{72490}-500}{12}$$

Divide -500-2\sqrt{72490} by 12.

$$x=-\frac{\sqrt{72490}}{6}-\frac{125}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{125}{3}+\frac{\sqrt{72490}}{6} for x_{1} and -\frac{125}{3}-\frac{\sqrt{72490}}{6} for x_{2}.

$$6x^{2}+500x-1665=6\left(x-\left(\frac{\sqrt{72490}}{6}-\frac{125}{3}\right)\right)\left(x-\left(-\frac{\sqrt{72490}}{6}-\frac{125}{3}\right)\right)$$

Evaluate

$$6x^{2}+500x-1665$$