Solve 6x^2-10x-6

$$6 { x }^{ 2 } -10x-6$$

Factor

$$6\left(x-\frac{5-\sqrt{61}}{6}\right)\left(x-\frac{\sqrt{61}+5}{6}\right)$$

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.

$$6x^{2}-10x-6=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 6\left(-6\right)}}{2\times 6}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 6\left(-6\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-10\right)±\sqrt{100-24\left(-6\right)}}{2\times 6}$$

Multiply -24 times -6.

$$x=\frac{-\left(-10\right)±\sqrt{100+144}}{2\times 6}$$

Add 100 to 144.

$$x=\frac{-\left(-10\right)±\sqrt{244}}{2\times 6}$$

Take the square root of 244.

$$x=\frac{-\left(-10\right)±2\sqrt{61}}{2\times 6}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{61}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{10±2\sqrt{61}}{12}$$

Now solve the equation x=\frac{10±2\sqrt{61}}{12} when ± is plus. Add 10 to 2\sqrt{61}.

$$x=\frac{2\sqrt{61}+10}{12}$$

Divide 10+2\sqrt{61} by 12.

$$x=\frac{\sqrt{61}+5}{6}$$

Now solve the equation x=\frac{10±2\sqrt{61}}{12} when ± is minus. Subtract 2\sqrt{61} from 10.

$$x=\frac{10-2\sqrt{61}}{12}$$

Divide 10-2\sqrt{61} by 12.

$$x=\frac{5-\sqrt{61}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{61}}{6} for x_{1} and \frac{5-\sqrt{61}}{6} for x_{2}.

$$6x^{2}-10x-6=6\left(x-\frac{\sqrt{61}+5}{6}\right)\left(x-\frac{5-\sqrt{61}}{6}\right)$$

Evaluate

$$6x^{2}-10x-6$$