Solve 3x^2-32x+72

$$3 { x }^{ 2 } -32x+72$$

Factor

$$3\left(x-\frac{16-2\sqrt{10}}{3}\right)\left(x-\frac{2\sqrt{10}+16}{3}\right)$$

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.

$$3x^{2}-32x+72=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 3\times 72}}{2\times 3}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 3\times 72}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-32\right)±\sqrt{1024-12\times 72}}{2\times 3}$$

Multiply -12 times 72.

$$x=\frac{-\left(-32\right)±\sqrt{1024-864}}{2\times 3}$$

Add 1024 to -864.

$$x=\frac{-\left(-32\right)±\sqrt{160}}{2\times 3}$$

Take the square root of 160.

$$x=\frac{-\left(-32\right)±4\sqrt{10}}{2\times 3}$$

The opposite of -32 is 32.

$$x=\frac{32±4\sqrt{10}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{32±4\sqrt{10}}{6}$$

Now solve the equation x=\frac{32±4\sqrt{10}}{6} when ± is plus. Add 32 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}+32}{6}$$

Divide 32+4\sqrt{10} by 6.

$$x=\frac{2\sqrt{10}+16}{3}$$

Now solve the equation x=\frac{32±4\sqrt{10}}{6} when ± is minus. Subtract 4\sqrt{10} from 32.

$$x=\frac{32-4\sqrt{10}}{6}$$

Divide 32-4\sqrt{10} by 6.

$$x=\frac{16-2\sqrt{10}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{16+2\sqrt{10}}{3} for x_{1} and \frac{16-2\sqrt{10}}{3} for x_{2}.

$$3x^{2}-32x+72=3\left(x-\frac{2\sqrt{10}+16}{3}\right)\left(x-\frac{16-2\sqrt{10}}{3}\right)$$

Evaluate

$$3x^{2}-32x+72$$