Solve 27x^2-66x+27

$$27 { x }^{ 2 } -66x+27$$

Factor

$$27\left(x-\frac{11-2\sqrt{10}}{9}\right)\left(x-\frac{2\sqrt{10}+11}{9}\right)$$

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.

$$27x^{2}-66x+27=0$$

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.

$$x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 27\times 27}}{2\times 27}$$

Square -66.

$$x=\frac{-\left(-66\right)±\sqrt{4356-4\times 27\times 27}}{2\times 27}$$

Multiply -4 times 27.

$$x=\frac{-\left(-66\right)±\sqrt{4356-108\times 27}}{2\times 27}$$

Multiply -108 times 27.

$$x=\frac{-\left(-66\right)±\sqrt{4356-2916}}{2\times 27}$$

Add 4356 to -2916.

$$x=\frac{-\left(-66\right)±\sqrt{1440}}{2\times 27}$$

Take the square root of 1440.

$$x=\frac{-\left(-66\right)±12\sqrt{10}}{2\times 27}$$

The opposite of -66 is 66.

$$x=\frac{66±12\sqrt{10}}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{66±12\sqrt{10}}{54}$$

Now solve the equation x=\frac{66±12\sqrt{10}}{54} when ± is plus. Add 66 to 12\sqrt{10}.

$$x=\frac{12\sqrt{10}+66}{54}$$

Divide 66+12\sqrt{10} by 54.

$$x=\frac{2\sqrt{10}+11}{9}$$

Now solve the equation x=\frac{66±12\sqrt{10}}{54} when ± is minus. Subtract 12\sqrt{10} from 66.

$$x=\frac{66-12\sqrt{10}}{54}$$

Divide 66-12\sqrt{10} by 54.

$$x=\frac{11-2\sqrt{10}}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+2\sqrt{10}}{9} for x_{1} and \frac{11-2\sqrt{10}}{9} for x_{2}.

$$27x^{2}-66x+27=27\left(x-\frac{2\sqrt{10}+11}{9}\right)\left(x-\frac{11-2\sqrt{10}}{9}\right)$$

Evaluate

$$27x^{2}-66x+27$$