Solve 45x^2-25x+3=

$$45 { x }^{ 2 } -25x+3=$$

Factor

$$45\left(x-\left(-\frac{\sqrt{85}}{90}+\frac{5}{18}\right)\right)\left(x-\left(\frac{\sqrt{85}}{90}+\frac{5}{18}\right)\right)$$

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.

$$45x^{2}-25x+3=0$$

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.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 45\times 3}}{2\times 45}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 45\times 3}}{2\times 45}$$

Multiply -4 times 45.

$$x=\frac{-\left(-25\right)±\sqrt{625-180\times 3}}{2\times 45}$$

Multiply -180 times 3.

$$x=\frac{-\left(-25\right)±\sqrt{625-540}}{2\times 45}$$

Add 625 to -540.

$$x=\frac{-\left(-25\right)±\sqrt{85}}{2\times 45}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{85}}{2\times 45}$$

Multiply 2 times 45.

$$x=\frac{25±\sqrt{85}}{90}$$

Now solve the equation x=\frac{25±\sqrt{85}}{90} when ± is plus. Add 25 to \sqrt{85}.

$$x=\frac{\sqrt{85}+25}{90}$$

Divide 25+\sqrt{85} by 90.

$$x=\frac{\sqrt{85}}{90}+\frac{5}{18}$$

Now solve the equation x=\frac{25±\sqrt{85}}{90} when ± is minus. Subtract \sqrt{85} from 25.

$$x=\frac{25-\sqrt{85}}{90}$$

Divide 25-\sqrt{85} by 90.

$$x=-\frac{\sqrt{85}}{90}+\frac{5}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{18}+\frac{\sqrt{85}}{90} for x_{1} and \frac{5}{18}-\frac{\sqrt{85}}{90} for x_{2}.

$$45x^{2}-25x+3=45\left(x-\left(\frac{\sqrt{85}}{90}+\frac{5}{18}\right)\right)\left(x-\left(-\frac{\sqrt{85}}{90}+\frac{5}{18}\right)\right)$$

Evaluate

$$45x^{2}-25x+3$$