Solve -4+20x-25x^2+77

$$- 4 + 20 x - 25 x ^ { 2 } + 77$$

Factor

$$-25\left(x-\frac{2-\sqrt{77}}{5}\right)\left(x-\frac{\sqrt{77}+2}{5}\right)$$

Steps Using the Quadratic Formula

Add -4 and 77 to get 73.

$$factor(73+20x-25x^{2})$$

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.

$$-25x^{2}+20x+73=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-25\right)\times 73}}{2\left(-25\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-25\right)\times 73}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-20±\sqrt{400+100\times 73}}{2\left(-25\right)}$$

Multiply 100 times 73.

$$x=\frac{-20±\sqrt{400+7300}}{2\left(-25\right)}$$

Add 400 to 7300.

$$x=\frac{-20±\sqrt{7700}}{2\left(-25\right)}$$

Take the square root of 7700.

$$x=\frac{-20±10\sqrt{77}}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{-20±10\sqrt{77}}{-50}$$

Now solve the equation x=\frac{-20±10\sqrt{77}}{-50} when ± is plus. Add -20 to 10\sqrt{77}.

$$x=\frac{10\sqrt{77}-20}{-50}$$

Divide -20+10\sqrt{77} by -50.

$$x=\frac{2-\sqrt{77}}{5}$$

Now solve the equation x=\frac{-20±10\sqrt{77}}{-50} when ± is minus. Subtract 10\sqrt{77} from -20.

$$x=\frac{-10\sqrt{77}-20}{-50}$$

Divide -20-10\sqrt{77} by -50.

$$x=\frac{\sqrt{77}+2}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2-\sqrt{77}}{5} for x_{1} and \frac{2+\sqrt{77}}{5} for x_{2}.

$$-25x^{2}+20x+73=-25\left(x-\frac{2-\sqrt{77}}{5}\right)\left(x-\frac{\sqrt{77}+2}{5}\right)$$

Evaluate

$$73+20x-25x^{2}$$

Solution Steps

Add -4 and 77 to get 73.

$$73+20x-25x^{2}$$