Solve 3x^2-15x+6

$$3 { x }^{ 2 } -15x+6$$

Factor

$$3\left(x-\frac{5-\sqrt{17}}{2}\right)\left(x-\frac{\sqrt{17}+5}{2}\right)$$

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.

$$3x^{2}-15x+6=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\times 6}}{2\times 3}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\times 6}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\times 6}}{2\times 3}$$

Multiply -12 times 6.

$$x=\frac{-\left(-15\right)±\sqrt{225-72}}{2\times 3}$$

Add 225 to -72.

$$x=\frac{-\left(-15\right)±\sqrt{153}}{2\times 3}$$

Take the square root of 153.

$$x=\frac{-\left(-15\right)±3\sqrt{17}}{2\times 3}$$

The opposite of -15 is 15.

$$x=\frac{15±3\sqrt{17}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{15±3\sqrt{17}}{6}$$

Now solve the equation x=\frac{15±3\sqrt{17}}{6} when ± is plus. Add 15 to 3\sqrt{17}.

$$x=\frac{3\sqrt{17}+15}{6}$$

Divide 15+3\sqrt{17} by 6.

$$x=\frac{\sqrt{17}+5}{2}$$

Now solve the equation x=\frac{15±3\sqrt{17}}{6} when ± is minus. Subtract 3\sqrt{17} from 15.

$$x=\frac{15-3\sqrt{17}}{6}$$

Divide 15-3\sqrt{17} by 6.

$$x=\frac{5-\sqrt{17}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{17}}{2} for x_{1} and \frac{5-\sqrt{17}}{2} for x_{2}.

$$3x^{2}-15x+6=3\left(x-\frac{\sqrt{17}+5}{2}\right)\left(x-\frac{5-\sqrt{17}}{2}\right)$$

Evaluate

$$3\left(x^{2}-5x+2\right)$$