Solve 3x^2-7x+3

$$3 { x }^{ 2 } -7x+3$$

Factor

$$3\left(x-\frac{7-\sqrt{13}}{6}\right)\left(x-\frac{\sqrt{13}+7}{6}\right)$$

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.

$$3x^{2}-7x+3=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\times 3}}{2\times 3}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\times 3}}{2\times 3}$$

Multiply -12 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-36}}{2\times 3}$$

Add 49 to -36.

$$x=\frac{-\left(-7\right)±\sqrt{13}}{2\times 3}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{7±\sqrt{13}}{6}$$

Now solve the equation x=\frac{7±\sqrt{13}}{6} when ± is plus. Add 7 to \sqrt{13}.

$$x=\frac{\sqrt{13}+7}{6}$$

Now solve the equation x=\frac{7±\sqrt{13}}{6} when ± is minus. Subtract \sqrt{13} from 7.

$$x=\frac{7-\sqrt{13}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{13}}{6} for x_{1} and \frac{7-\sqrt{13}}{6} for x_{2}.

$$3x^{2}-7x+3=3\left(x-\frac{\sqrt{13}+7}{6}\right)\left(x-\frac{7-\sqrt{13}}{6}\right)$$

Evaluate

$$3x^{2}-7x+3$$