Solve -6x^2+33x+15

$$-6 { x }^{ 2 } +33x+15$$

Factor

$$-6\left(x-\frac{11-\sqrt{161}}{4}\right)\left(x-\frac{\sqrt{161}+11}{4}\right)$$

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.

$$-6x^{2}+33x+15=0$$

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.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-6\right)\times 15}}{2\left(-6\right)}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\left(-6\right)\times 15}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-33±\sqrt{1089+24\times 15}}{2\left(-6\right)}$$

Multiply 24 times 15.

$$x=\frac{-33±\sqrt{1089+360}}{2\left(-6\right)}$$

Add 1089 to 360.

$$x=\frac{-33±\sqrt{1449}}{2\left(-6\right)}$$

Take the square root of 1449.

$$x=\frac{-33±3\sqrt{161}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-33±3\sqrt{161}}{-12}$$

Now solve the equation x=\frac{-33±3\sqrt{161}}{-12} when ± is plus. Add -33 to 3\sqrt{161}.

$$x=\frac{3\sqrt{161}-33}{-12}$$

Divide -33+3\sqrt{161} by -12.

$$x=\frac{11-\sqrt{161}}{4}$$

Now solve the equation x=\frac{-33±3\sqrt{161}}{-12} when ± is minus. Subtract 3\sqrt{161} from -33.

$$x=\frac{-3\sqrt{161}-33}{-12}$$

Divide -33-3\sqrt{161} by -12.

$$x=\frac{\sqrt{161}+11}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11-\sqrt{161}}{4} for x_{1} and \frac{11+\sqrt{161}}{4} for x_{2}.

$$-6x^{2}+33x+15=-6\left(x-\frac{11-\sqrt{161}}{4}\right)\left(x-\frac{\sqrt{161}+11}{4}\right)$$

Evaluate

$$15+33x-6x^{2}$$