Solve 2x^2-12x-7

$$2 { x }^{ 2 } -12x-7$$

Factor

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

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.

$$2x^{2}-12x-7=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-12\right)±\sqrt{144-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-12\right)±\sqrt{144+56}}{2\times 2}$$

Add 144 to 56.

$$x=\frac{-\left(-12\right)±\sqrt{200}}{2\times 2}$$

Take the square root of 200.

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

The opposite of -12 is 12.

$$x=\frac{12±10\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{12±10\sqrt{2}}{4}$$

Now solve the equation x=\frac{12±10\sqrt{2}}{4} when ± is plus. Add 12 to 10\sqrt{2}.

$$x=\frac{10\sqrt{2}+12}{4}$$

Divide 12+10\sqrt{2} by 4.

$$x=\frac{5\sqrt{2}}{2}+3$$

Now solve the equation x=\frac{12±10\sqrt{2}}{4} when ± is minus. Subtract 10\sqrt{2} from 12.

$$x=\frac{12-10\sqrt{2}}{4}$$

Divide 12-10\sqrt{2} by 4.

$$x=-\frac{5\sqrt{2}}{2}+3$$

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

$$2x^{2}-12x-7=2\left(x-\left(\frac{5\sqrt{2}}{2}+3\right)\right)\left(x-\left(-\frac{5\sqrt{2}}{2}+3\right)\right)$$

Evaluate

$$2x^{2}-12x-7$$