Solve 4x^2-12x-11

$$4 { x }^{ 2 } -12x-11$$

Factor

$$4\left(x-\left(\frac{3}{2}-\sqrt{5}\right)\right)\left(x-\left(\sqrt{5}+\frac{3}{2}\right)\right)$$

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.

$$4x^{2}-12x-11=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-11\right)}}{2\times 4}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-11\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-11\right)}}{2\times 4}$$

Multiply -16 times -11.

$$x=\frac{-\left(-12\right)±\sqrt{144+176}}{2\times 4}$$

Add 144 to 176.

$$x=\frac{-\left(-12\right)±\sqrt{320}}{2\times 4}$$

Take the square root of 320.

$$x=\frac{-\left(-12\right)±8\sqrt{5}}{2\times 4}$$

The opposite of -12 is 12.

$$x=\frac{12±8\sqrt{5}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{12±8\sqrt{5}}{8}$$

Now solve the equation x=\frac{12±8\sqrt{5}}{8} when ± is plus. Add 12 to 8\sqrt{5}.

$$x=\frac{8\sqrt{5}+12}{8}$$

Divide 12+8\sqrt{5} by 8.

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

Now solve the equation x=\frac{12±8\sqrt{5}}{8} when ± is minus. Subtract 8\sqrt{5} from 12.

$$x=\frac{12-8\sqrt{5}}{8}$$

Divide 12-8\sqrt{5} by 8.

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

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

$$4x^{2}-12x-11=4\left(x-\left(\sqrt{5}+\frac{3}{2}\right)\right)\left(x-\left(\frac{3}{2}-\sqrt{5}\right)\right)$$

Evaluate

$$4x^{2}-12x-11$$