Solve 45x^2-255x+2

$$45 { x }^{ 2 } -255x+2$$

Factor

$$45\left(x-\left(-\frac{\sqrt{7185}}{30}+\frac{17}{6}\right)\right)\left(x-\left(\frac{\sqrt{7185}}{30}+\frac{17}{6}\right)\right)$$

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.

$$45x^{2}-255x+2=0$$

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.

$$x=\frac{-\left(-255\right)±\sqrt{\left(-255\right)^{2}-4\times 45\times 2}}{2\times 45}$$

Square -255.

$$x=\frac{-\left(-255\right)±\sqrt{65025-4\times 45\times 2}}{2\times 45}$$

Multiply -4 times 45.

$$x=\frac{-\left(-255\right)±\sqrt{65025-180\times 2}}{2\times 45}$$

Multiply -180 times 2.

$$x=\frac{-\left(-255\right)±\sqrt{65025-360}}{2\times 45}$$

Add 65025 to -360.

$$x=\frac{-\left(-255\right)±\sqrt{64665}}{2\times 45}$$

Take the square root of 64665.

$$x=\frac{-\left(-255\right)±3\sqrt{7185}}{2\times 45}$$

The opposite of -255 is 255.

$$x=\frac{255±3\sqrt{7185}}{2\times 45}$$

Multiply 2 times 45.

$$x=\frac{255±3\sqrt{7185}}{90}$$

Now solve the equation x=\frac{255±3\sqrt{7185}}{90} when ± is plus. Add 255 to 3\sqrt{7185}.

$$x=\frac{3\sqrt{7185}+255}{90}$$

Divide 255+3\sqrt{7185} by 90.

$$x=\frac{\sqrt{7185}}{30}+\frac{17}{6}$$

Now solve the equation x=\frac{255±3\sqrt{7185}}{90} when ± is minus. Subtract 3\sqrt{7185} from 255.

$$x=\frac{255-3\sqrt{7185}}{90}$$

Divide 255-3\sqrt{7185} by 90.

$$x=-\frac{\sqrt{7185}}{30}+\frac{17}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{17}{6}+\frac{\sqrt{7185}}{30} for x_{1} and \frac{17}{6}-\frac{\sqrt{7185}}{30} for x_{2}.

$$45x^{2}-255x+2=45\left(x-\left(\frac{\sqrt{7185}}{30}+\frac{17}{6}\right)\right)\left(x-\left(-\frac{\sqrt{7185}}{30}+\frac{17}{6}\right)\right)$$

Evaluate

$$45x^{2}-255x+2$$