Solve 10x^2-12x-4

$$10 { x }^{ 2 } -12x-4$$

Factor

$$10\left(x-\frac{3-\sqrt{19}}{5}\right)\left(x-\frac{\sqrt{19}+3}{5}\right)$$

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.

$$10x^{2}-12x-4=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 10\left(-4\right)}}{2\times 10}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 10\left(-4\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-12\right)±\sqrt{144-40\left(-4\right)}}{2\times 10}$$

Multiply -40 times -4.

$$x=\frac{-\left(-12\right)±\sqrt{144+160}}{2\times 10}$$

Add 144 to 160.

$$x=\frac{-\left(-12\right)±\sqrt{304}}{2\times 10}$$

Take the square root of 304.

$$x=\frac{-\left(-12\right)±4\sqrt{19}}{2\times 10}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{19}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{12±4\sqrt{19}}{20}$$

Now solve the equation x=\frac{12±4\sqrt{19}}{20} when ± is plus. Add 12 to 4\sqrt{19}.

$$x=\frac{4\sqrt{19}+12}{20}$$

Divide 12+4\sqrt{19} by 20.

$$x=\frac{\sqrt{19}+3}{5}$$

Now solve the equation x=\frac{12±4\sqrt{19}}{20} when ± is minus. Subtract 4\sqrt{19} from 12.

$$x=\frac{12-4\sqrt{19}}{20}$$

Divide 12-4\sqrt{19} by 20.

$$x=\frac{3-\sqrt{19}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{19}}{5} for x_{1} and \frac{3-\sqrt{19}}{5} for x_{2}.

$$10x^{2}-12x-4=10\left(x-\frac{\sqrt{19}+3}{5}\right)\left(x-\frac{3-\sqrt{19}}{5}\right)$$

Evaluate

$$10x^{2}-12x-4$$