Solve (-11x^2+2x-4)+(14x^2+5x-9)

$$( - 11 x ^ { 2 } + 2 x - 4 ) + ( 14 x ^ { 2 } + 5 x - 9 )$$

Evaluate

$$3x^{2}+7x-13$$

Solution Steps

Combine -11x^{2} and 14x^{2} to get 3x^{2}.

$$3x^{2}+2x-4+5x-9$$

Combine 2x and 5x to get 7x.

$$3x^{2}+7x-4-9$$

Subtract 9 from -4 to get -13.

$$3x^{2}+7x-13$$

Factor

$$3\left(x-\frac{-\sqrt{205}-7}{6}\right)\left(x-\frac{\sqrt{205}-7}{6}\right)$$

Steps Using the Quadratic Formula

Combine -11x^{2} and 14x^{2} to get 3x^{2}.

$$factor(3x^{2}+2x-4+5x-9)$$

Combine 2x and 5x to get 7x.

$$factor(3x^{2}+7x-4-9)$$

Subtract 9 from -4 to get -13.

$$factor(3x^{2}+7x-13)$$

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.

$$3x^{2}+7x-13=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\left(-13\right)}}{2\times 3}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 3\left(-13\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-7±\sqrt{49-12\left(-13\right)}}{2\times 3}$$

Multiply -12 times -13.

$$x=\frac{-7±\sqrt{49+156}}{2\times 3}$$

Add 49 to 156.

$$x=\frac{-7±\sqrt{205}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-7±\sqrt{205}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{205}}{6} when ± is plus. Add -7 to \sqrt{205}.

$$x=\frac{\sqrt{205}-7}{6}$$

Now solve the equation x=\frac{-7±\sqrt{205}}{6} when ± is minus. Subtract \sqrt{205} from -7.

$$x=\frac{-\sqrt{205}-7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{205}}{6} for x_{1} and \frac{-7-\sqrt{205}}{6} for x_{2}.

$$3x^{2}+7x-13=3\left(x-\frac{\sqrt{205}-7}{6}\right)\left(x-\frac{-\sqrt{205}-7}{6}\right)$$