Solve {l}{(4x^2+9x+2)}{+(-2x^2+2x-3)}

$$\left. \begin{array} { l } { ( 4 x ^ { 2 } + 9 x + 2 ) } \\ { + ( - 2 x ^ { 2 } + 2 x - 3 ) } \end{array} \right.$$

Evaluate

$$2x^{2}+11x-1$$

Solution Steps

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}+9x+2+2x-3$$

Combine 9x and 2x to get 11x.

$$2x^{2}+11x+2-3$$

Subtract 3 from 2 to get -1.

$$2x^{2}+11x-1$$

Factor

$$2\left(x-\frac{-\sqrt{129}-11}{4}\right)\left(x-\frac{\sqrt{129}-11}{4}\right)$$

Steps Using the Quadratic Formula

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$factor(2x^{2}+9x+2+2x-3)$$

Combine 9x and 2x to get 11x.

$$factor(2x^{2}+11x+2-3)$$

Subtract 3 from 2 to get -1.

$$factor(2x^{2}+11x-1)$$

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.

$$2x^{2}+11x-1=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 2\left(-1\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-11±\sqrt{121-8\left(-1\right)}}{2\times 2}$$

Multiply -8 times -1.

$$x=\frac{-11±\sqrt{121+8}}{2\times 2}$$

Add 121 to 8.

$$x=\frac{-11±\sqrt{129}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-11±\sqrt{129}}{4}$$

Now solve the equation x=\frac{-11±\sqrt{129}}{4} when ± is plus. Add -11 to \sqrt{129}.

$$x=\frac{\sqrt{129}-11}{4}$$

Now solve the equation x=\frac{-11±\sqrt{129}}{4} when ± is minus. Subtract \sqrt{129} from -11.

$$x=\frac{-\sqrt{129}-11}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-11+\sqrt{129}}{4} for x_{1} and \frac{-11-\sqrt{129}}{4} for x_{2}.

$$2x^{2}+11x-1=2\left(x-\frac{\sqrt{129}-11}{4}\right)\left(x-\frac{-\sqrt{129}-11}{4}\right)$$