Solve -40x^2+106x-40

$$-40 { x }^{ 2 } +106x-40$$

Factor

$$-40\left(x-\frac{53-\sqrt{1209}}{40}\right)\left(x-\frac{\sqrt{1209}+53}{40}\right)$$

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.

$$-40x^{2}+106x-40=0$$

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.

$$x=\frac{-106±\sqrt{106^{2}-4\left(-40\right)\left(-40\right)}}{2\left(-40\right)}$$

Square 106.

$$x=\frac{-106±\sqrt{11236-4\left(-40\right)\left(-40\right)}}{2\left(-40\right)}$$

Multiply -4 times -40.

$$x=\frac{-106±\sqrt{11236+160\left(-40\right)}}{2\left(-40\right)}$$

Multiply 160 times -40.

$$x=\frac{-106±\sqrt{11236-6400}}{2\left(-40\right)}$$

Add 11236 to -6400.

$$x=\frac{-106±\sqrt{4836}}{2\left(-40\right)}$$

Take the square root of 4836.

$$x=\frac{-106±2\sqrt{1209}}{2\left(-40\right)}$$

Multiply 2 times -40.

$$x=\frac{-106±2\sqrt{1209}}{-80}$$

Now solve the equation x=\frac{-106±2\sqrt{1209}}{-80} when ± is plus. Add -106 to 2\sqrt{1209}.

$$x=\frac{2\sqrt{1209}-106}{-80}$$

Divide -106+2\sqrt{1209} by -80.

$$x=\frac{53-\sqrt{1209}}{40}$$

Now solve the equation x=\frac{-106±2\sqrt{1209}}{-80} when ± is minus. Subtract 2\sqrt{1209} from -106.

$$x=\frac{-2\sqrt{1209}-106}{-80}$$

Divide -106-2\sqrt{1209} by -80.

$$x=\frac{\sqrt{1209}+53}{40}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{53-\sqrt{1209}}{40} for x_{1} and \frac{53+\sqrt{1209}}{40} for x_{2}.

$$-40x^{2}+106x-40=-40\left(x-\frac{53-\sqrt{1209}}{40}\right)\left(x-\frac{\sqrt{1209}+53}{40}\right)$$

Evaluate

$$-40x^{2}+106x-40$$