Solve 5x^2-12x+3

$$5 { x }^{ 2 } -12x+3$$

Factor

$$5\left(x-\frac{6-\sqrt{21}}{5}\right)\left(x-\frac{\sqrt{21}+6}{5}\right)$$

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.

$$5x^{2}-12x+3=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 5\times 3}}{2\times 5}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 5\times 3}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-12\right)±\sqrt{144-20\times 3}}{2\times 5}$$

Multiply -20 times 3.

$$x=\frac{-\left(-12\right)±\sqrt{144-60}}{2\times 5}$$

Add 144 to -60.

$$x=\frac{-\left(-12\right)±\sqrt{84}}{2\times 5}$$

Take the square root of 84.

$$x=\frac{-\left(-12\right)±2\sqrt{21}}{2\times 5}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{21}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{12±2\sqrt{21}}{10}$$

Now solve the equation x=\frac{12±2\sqrt{21}}{10} when ± is plus. Add 12 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}+12}{10}$$

Divide 12+2\sqrt{21} by 10.

$$x=\frac{\sqrt{21}+6}{5}$$

Now solve the equation x=\frac{12±2\sqrt{21}}{10} when ± is minus. Subtract 2\sqrt{21} from 12.

$$x=\frac{12-2\sqrt{21}}{10}$$

Divide 12-2\sqrt{21} by 10.

$$x=\frac{6-\sqrt{21}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{6+\sqrt{21}}{5} for x_{1} and \frac{6-\sqrt{21}}{5} for x_{2}.

$$5x^{2}-12x+3=5\left(x-\frac{\sqrt{21}+6}{5}\right)\left(x-\frac{6-\sqrt{21}}{5}\right)$$

Evaluate

$$5x^{2}-12x+3$$