Solve -3x^2+117x-900

$$-3 { x }^{ 2 } +117x-900$$

Factor

$$-3\left(x-\frac{39-\sqrt{321}}{2}\right)\left(x-\frac{\sqrt{321}+39}{2}\right)$$

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.

$$-3x^{2}+117x-900=0$$

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.

$$x=\frac{-117±\sqrt{117^{2}-4\left(-3\right)\left(-900\right)}}{2\left(-3\right)}$$

Square 117.

$$x=\frac{-117±\sqrt{13689-4\left(-3\right)\left(-900\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-117±\sqrt{13689+12\left(-900\right)}}{2\left(-3\right)}$$

Multiply 12 times -900.

$$x=\frac{-117±\sqrt{13689-10800}}{2\left(-3\right)}$$

Add 13689 to -10800.

$$x=\frac{-117±\sqrt{2889}}{2\left(-3\right)}$$

Take the square root of 2889.

$$x=\frac{-117±3\sqrt{321}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-117±3\sqrt{321}}{-6}$$

Now solve the equation x=\frac{-117±3\sqrt{321}}{-6} when ± is plus. Add -117 to 3\sqrt{321}.

$$x=\frac{3\sqrt{321}-117}{-6}$$

Divide -117+3\sqrt{321} by -6.

$$x=\frac{39-\sqrt{321}}{2}$$

Now solve the equation x=\frac{-117±3\sqrt{321}}{-6} when ± is minus. Subtract 3\sqrt{321} from -117.

$$x=\frac{-3\sqrt{321}-117}{-6}$$

Divide -117-3\sqrt{321} by -6.

$$x=\frac{\sqrt{321}+39}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{39-\sqrt{321}}{2} for x_{1} and \frac{39+\sqrt{321}}{2} for x_{2}.

$$-3x^{2}+117x-900=-3\left(x-\frac{39-\sqrt{321}}{2}\right)\left(x-\frac{\sqrt{321}+39}{2}\right)$$

Evaluate

$$-3x^{2}+117x-900$$