Solve x^2+5x-65+x^2-2

$$x ^ { 2 } + 5 x - 65 + x ^ { 2 } - 2$$

Evaluate

$$2x^{2}+5x-67$$

Solution Steps

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+5x-65-2$$

Subtract 2 from -65 to get -67.

$$2x^{2}+5x-67$$

Factor

$$2\left(x-\frac{-\sqrt{561}-5}{4}\right)\left(x-\frac{\sqrt{561}-5}{4}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and x^{2} to get 2x^{2}.

$$factor(2x^{2}+5x-65-2)$$

Subtract 2 from -65 to get -67.

$$factor(2x^{2}+5x-67)$$

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.

$$2x^{2}+5x-67=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-67\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-67\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-67\right)}}{2\times 2}$$

Multiply -8 times -67.

$$x=\frac{-5±\sqrt{25+536}}{2\times 2}$$

Add 25 to 536.

$$x=\frac{-5±\sqrt{561}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±\sqrt{561}}{4}$$

Now solve the equation x=\frac{-5±\sqrt{561}}{4} when ± is plus. Add -5 to \sqrt{561}.

$$x=\frac{\sqrt{561}-5}{4}$$

Now solve the equation x=\frac{-5±\sqrt{561}}{4} when ± is minus. Subtract \sqrt{561} from -5.

$$x=\frac{-\sqrt{561}-5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{561}}{4} for x_{1} and \frac{-5-\sqrt{561}}{4} for x_{2}.

$$2x^{2}+5x-67=2\left(x-\frac{\sqrt{561}-5}{4}\right)\left(x-\frac{-\sqrt{561}-5}{4}\right)$$