Solve 6x^2-57x+45

$$6 { x }^{ 2 } -57x+45$$

Factor

$$6\left(x-\frac{19-\sqrt{241}}{4}\right)\left(x-\frac{\sqrt{241}+19}{4}\right)$$

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.

$$6x^{2}-57x+45=0$$

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.

$$x=\frac{-\left(-57\right)±\sqrt{\left(-57\right)^{2}-4\times 6\times 45}}{2\times 6}$$

Square -57.

$$x=\frac{-\left(-57\right)±\sqrt{3249-4\times 6\times 45}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-57\right)±\sqrt{3249-24\times 45}}{2\times 6}$$

Multiply -24 times 45.

$$x=\frac{-\left(-57\right)±\sqrt{3249-1080}}{2\times 6}$$

Add 3249 to -1080.

$$x=\frac{-\left(-57\right)±\sqrt{2169}}{2\times 6}$$

Take the square root of 2169.

$$x=\frac{-\left(-57\right)±3\sqrt{241}}{2\times 6}$$

The opposite of -57 is 57.

$$x=\frac{57±3\sqrt{241}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{57±3\sqrt{241}}{12}$$

Now solve the equation x=\frac{57±3\sqrt{241}}{12} when ± is plus. Add 57 to 3\sqrt{241}.

$$x=\frac{3\sqrt{241}+57}{12}$$

Divide 57+3\sqrt{241} by 12.

$$x=\frac{\sqrt{241}+19}{4}$$

Now solve the equation x=\frac{57±3\sqrt{241}}{12} when ± is minus. Subtract 3\sqrt{241} from 57.

$$x=\frac{57-3\sqrt{241}}{12}$$

Divide 57-3\sqrt{241} by 12.

$$x=\frac{19-\sqrt{241}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19+\sqrt{241}}{4} for x_{1} and \frac{19-\sqrt{241}}{4} for x_{2}.

$$6x^{2}-57x+45=6\left(x-\frac{\sqrt{241}+19}{4}\right)\left(x-\frac{19-\sqrt{241}}{4}\right)$$

Evaluate

$$6x^{2}-57x+45$$