Solve 2x^2-5x-12^2

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

Factor

$$2\left(x-\frac{5-\sqrt{1177}}{4}\right)\left(x-\frac{\sqrt{1177}+5}{4}\right)$$

Steps Using the Quadratic Formula

Calculate 12 to the power of 2 and get 144.

$$factor(2x^{2}-5x-144)$$

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.

$$2x^{2}-5x-144=0$$

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.

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

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-144\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\left(-144\right)}}{2\times 2}$$

Multiply -8 times -144.

$$x=\frac{-\left(-5\right)±\sqrt{25+1152}}{2\times 2}$$

Add 25 to 1152.

$$x=\frac{-\left(-5\right)±\sqrt{1177}}{2\times 2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{1177}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{5±\sqrt{1177}}{4}$$

Now solve the equation x=\frac{5±\sqrt{1177}}{4} when ± is plus. Add 5 to \sqrt{1177}.

$$x=\frac{\sqrt{1177}+5}{4}$$

Now solve the equation x=\frac{5±\sqrt{1177}}{4} when ± is minus. Subtract \sqrt{1177} from 5.

$$x=\frac{5-\sqrt{1177}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{1177}}{4} for x_{1} and \frac{5-\sqrt{1177}}{4} for x_{2}.

$$2x^{2}-5x-144=2\left(x-\frac{\sqrt{1177}+5}{4}\right)\left(x-\frac{5-\sqrt{1177}}{4}\right)$$

Evaluate

$$2x^{2}-5x-144$$

Solution Steps

Calculate 12 to the power of 2 and get 144.

$$2x^{2}-5x-144$$