Solve -80x^2+2600x-2000

$$-80 { x }^{ 2 } +2600x-2000$$

Factor

$$-80\left(x-\frac{65-15\sqrt{17}}{4}\right)\left(x-\frac{15\sqrt{17}+65}{4}\right)$$

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.

$$-80x^{2}+2600x-2000=0$$

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.

$$x=\frac{-2600±\sqrt{2600^{2}-4\left(-80\right)\left(-2000\right)}}{2\left(-80\right)}$$

Square 2600.

$$x=\frac{-2600±\sqrt{6760000-4\left(-80\right)\left(-2000\right)}}{2\left(-80\right)}$$

Multiply -4 times -80.

$$x=\frac{-2600±\sqrt{6760000+320\left(-2000\right)}}{2\left(-80\right)}$$

Multiply 320 times -2000.

$$x=\frac{-2600±\sqrt{6760000-640000}}{2\left(-80\right)}$$

Add 6760000 to -640000.

$$x=\frac{-2600±\sqrt{6120000}}{2\left(-80\right)}$$

Take the square root of 6120000.

$$x=\frac{-2600±600\sqrt{17}}{2\left(-80\right)}$$

Multiply 2 times -80.

$$x=\frac{-2600±600\sqrt{17}}{-160}$$

Now solve the equation x=\frac{-2600±600\sqrt{17}}{-160} when ± is plus. Add -2600 to 600\sqrt{17}.

$$x=\frac{600\sqrt{17}-2600}{-160}$$

Divide -2600+600\sqrt{17} by -160.

$$x=\frac{65-15\sqrt{17}}{4}$$

Now solve the equation x=\frac{-2600±600\sqrt{17}}{-160} when ± is minus. Subtract 600\sqrt{17} from -2600.

$$x=\frac{-600\sqrt{17}-2600}{-160}$$

Divide -2600-600\sqrt{17} by -160.

$$x=\frac{15\sqrt{17}+65}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{65-15\sqrt{17}}{4} for x_{1} and \frac{65+15\sqrt{17}}{4} for x_{2}.

$$-80x^{2}+2600x-2000=-80\left(x-\frac{65-15\sqrt{17}}{4}\right)\left(x-\frac{15\sqrt{17}+65}{4}\right)$$

Evaluate

$$-80x^{2}+2600x-2000$$