Solve 272+60x-18x^2

$$272+60x-18 { x }^{ 2 }$$

Factor

$$-18\left(x-\frac{5-\sqrt{161}}{3}\right)\left(x-\frac{\sqrt{161}+5}{3}\right)$$

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.

$$-18x^{2}+60x+272=0$$

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.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-18\right)\times 272}}{2\left(-18\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-18\right)\times 272}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-60±\sqrt{3600+72\times 272}}{2\left(-18\right)}$$

Multiply 72 times 272.

$$x=\frac{-60±\sqrt{3600+19584}}{2\left(-18\right)}$$

Add 3600 to 19584.

$$x=\frac{-60±\sqrt{23184}}{2\left(-18\right)}$$

Take the square root of 23184.

$$x=\frac{-60±12\sqrt{161}}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-60±12\sqrt{161}}{-36}$$

Now solve the equation x=\frac{-60±12\sqrt{161}}{-36} when ± is plus. Add -60 to 12\sqrt{161}.

$$x=\frac{12\sqrt{161}-60}{-36}$$

Divide -60+12\sqrt{161} by -36.

$$x=\frac{5-\sqrt{161}}{3}$$

Now solve the equation x=\frac{-60±12\sqrt{161}}{-36} when ± is minus. Subtract 12\sqrt{161} from -60.

$$x=\frac{-12\sqrt{161}-60}{-36}$$

Divide -60-12\sqrt{161} by -36.

$$x=\frac{\sqrt{161}+5}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{161}}{3} for x_{1} and \frac{5+\sqrt{161}}{3} for x_{2}.

$$-18x^{2}+60x+272=-18\left(x-\frac{5-\sqrt{161}}{3}\right)\left(x-\frac{\sqrt{161}+5}{3}\right)$$

Evaluate

$$272+60x-18x^{2}$$