Solve 9x^2-36x+21

$$9 { x }^{ 2 } -36x+21$$

Factor

$$9\left(x-\left(-\frac{\sqrt{15}}{3}+2\right)\right)\left(x-\left(\frac{\sqrt{15}}{3}+2\right)\right)$$

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.

$$9x^{2}-36x+21=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 9\times 21}}{2\times 9}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 9\times 21}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-36\right)±\sqrt{1296-36\times 21}}{2\times 9}$$

Multiply -36 times 21.

$$x=\frac{-\left(-36\right)±\sqrt{1296-756}}{2\times 9}$$

Add 1296 to -756.

$$x=\frac{-\left(-36\right)±\sqrt{540}}{2\times 9}$$

Take the square root of 540.

$$x=\frac{-\left(-36\right)±6\sqrt{15}}{2\times 9}$$

The opposite of -36 is 36.

$$x=\frac{36±6\sqrt{15}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{36±6\sqrt{15}}{18}$$

Now solve the equation x=\frac{36±6\sqrt{15}}{18} when ± is plus. Add 36 to 6\sqrt{15}.

$$x=\frac{6\sqrt{15}+36}{18}$$

Divide 36+6\sqrt{15} by 18.

$$x=\frac{\sqrt{15}}{3}+2$$

Now solve the equation x=\frac{36±6\sqrt{15}}{18} when ± is minus. Subtract 6\sqrt{15} from 36.

$$x=\frac{36-6\sqrt{15}}{18}$$

Divide 36-6\sqrt{15} by 18.

$$x=-\frac{\sqrt{15}}{3}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2+\frac{\sqrt{15}}{3} for x_{1} and 2-\frac{\sqrt{15}}{3} for x_{2}.

$$9x^{2}-36x+21=9\left(x-\left(\frac{\sqrt{15}}{3}+2\right)\right)\left(x-\left(-\frac{\sqrt{15}}{3}+2\right)\right)$$

Evaluate

$$9x^{2}-36x+21$$