Solve (7x^2+11x-6)+(10x^2+14x+9)

$$(7 { x }^{ 2 } +11x-6)+(10 { x }^{ 2 } +14x+9)$$

Evaluate

$$17x^{2}+25x+3$$

Solution Steps

Combine 7x^{2} and 10x^{2} to get 17x^{2}.

$$17x^{2}+11x-6+14x+9$$

Combine 11x and 14x to get 25x.

$$17x^{2}+25x-6+9$$

Add -6 and 9 to get 3.

$$17x^{2}+25x+3$$

Factor

$$17\left(x-\frac{-\sqrt{421}-25}{34}\right)\left(x-\frac{\sqrt{421}-25}{34}\right)$$

Steps Using the Quadratic Formula

Combine 7x^{2} and 10x^{2} to get 17x^{2}.

$$factor(17x^{2}+11x-6+14x+9)$$

Combine 11x and 14x to get 25x.

$$factor(17x^{2}+25x-6+9)$$

Add -6 and 9 to get 3.

$$factor(17x^{2}+25x+3)$$

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.

$$17x^{2}+25x+3=0$$

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.

$$x=\frac{-25±\sqrt{25^{2}-4\times 17\times 3}}{2\times 17}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 17\times 3}}{2\times 17}$$

Multiply -4 times 17.

$$x=\frac{-25±\sqrt{625-68\times 3}}{2\times 17}$$

Multiply -68 times 3.

$$x=\frac{-25±\sqrt{625-204}}{2\times 17}$$

Add 625 to -204.

$$x=\frac{-25±\sqrt{421}}{2\times 17}$$

Multiply 2 times 17.

$$x=\frac{-25±\sqrt{421}}{34}$$

Now solve the equation x=\frac{-25±\sqrt{421}}{34} when ± is plus. Add -25 to \sqrt{421}.

$$x=\frac{\sqrt{421}-25}{34}$$

Now solve the equation x=\frac{-25±\sqrt{421}}{34} when ± is minus. Subtract \sqrt{421} from -25.

$$x=\frac{-\sqrt{421}-25}{34}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-25+\sqrt{421}}{34} for x_{1} and \frac{-25-\sqrt{421}}{34} for x_{2}.

$$17x^{2}+25x+3=17\left(x-\frac{\sqrt{421}-25}{34}\right)\left(x-\frac{-\sqrt{421}-25}{34}\right)$$