Solve -20x^2+66x-20

$$-20 { x }^{ 2 } +66x-20$$

Factor

$$-20\left(x-\frac{33-\sqrt{689}}{20}\right)\left(x-\frac{\sqrt{689}+33}{20}\right)$$

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.

$$-20x^{2}+66x-20=0$$

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.

$$x=\frac{-66±\sqrt{66^{2}-4\left(-20\right)\left(-20\right)}}{2\left(-20\right)}$$

Square 66.

$$x=\frac{-66±\sqrt{4356-4\left(-20\right)\left(-20\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-66±\sqrt{4356+80\left(-20\right)}}{2\left(-20\right)}$$

Multiply 80 times -20.

$$x=\frac{-66±\sqrt{4356-1600}}{2\left(-20\right)}$$

Add 4356 to -1600.

$$x=\frac{-66±\sqrt{2756}}{2\left(-20\right)}$$

Take the square root of 2756.

$$x=\frac{-66±2\sqrt{689}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-66±2\sqrt{689}}{-40}$$

Now solve the equation x=\frac{-66±2\sqrt{689}}{-40} when ± is plus. Add -66 to 2\sqrt{689}.

$$x=\frac{2\sqrt{689}-66}{-40}$$

Divide -66+2\sqrt{689} by -40.

$$x=\frac{33-\sqrt{689}}{20}$$

Now solve the equation x=\frac{-66±2\sqrt{689}}{-40} when ± is minus. Subtract 2\sqrt{689} from -66.

$$x=\frac{-2\sqrt{689}-66}{-40}$$

Divide -66-2\sqrt{689} by -40.

$$x=\frac{\sqrt{689}+33}{20}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{33-\sqrt{689}}{20} for x_{1} and \frac{33+\sqrt{689}}{20} for x_{2}.

$$-20x^{2}+66x-20=-20\left(x-\frac{33-\sqrt{689}}{20}\right)\left(x-\frac{\sqrt{689}+33}{20}\right)$$

Evaluate

$$-20x^{2}+66x-20$$