Solve 9x^2-200x-455

$$9 { x }^{ 2 } -200x-455$$

Factor

$$9\left(x-\frac{100-\sqrt{14095}}{9}\right)\left(x-\frac{\sqrt{14095}+100}{9}\right)$$

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.

$$9x^{2}-200x-455=0$$

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.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\times 9\left(-455\right)}}{2\times 9}$$

Square -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\times 9\left(-455\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-200\right)±\sqrt{40000-36\left(-455\right)}}{2\times 9}$$

Multiply -36 times -455.

$$x=\frac{-\left(-200\right)±\sqrt{40000+16380}}{2\times 9}$$

Add 40000 to 16380.

$$x=\frac{-\left(-200\right)±\sqrt{56380}}{2\times 9}$$

Take the square root of 56380.

$$x=\frac{-\left(-200\right)±2\sqrt{14095}}{2\times 9}$$

The opposite of -200 is 200.

$$x=\frac{200±2\sqrt{14095}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{200±2\sqrt{14095}}{18}$$

Now solve the equation x=\frac{200±2\sqrt{14095}}{18} when ± is plus. Add 200 to 2\sqrt{14095}.

$$x=\frac{2\sqrt{14095}+200}{18}$$

Divide 200+2\sqrt{14095} by 18.

$$x=\frac{\sqrt{14095}+100}{9}$$

Now solve the equation x=\frac{200±2\sqrt{14095}}{18} when ± is minus. Subtract 2\sqrt{14095} from 200.

$$x=\frac{200-2\sqrt{14095}}{18}$$

Divide 200-2\sqrt{14095} by 18.

$$x=\frac{100-\sqrt{14095}}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{100+\sqrt{14095}}{9} for x_{1} and \frac{100-\sqrt{14095}}{9} for x_{2}.

$$9x^{2}-200x-455=9\left(x-\frac{\sqrt{14095}+100}{9}\right)\left(x-\frac{100-\sqrt{14095}}{9}\right)$$

Evaluate

$$9x^{2}-200x-455$$