Solve (-3x^2-9x+5)+(-9x^2+6)

$$( - 3 x ^ { 2 } - 9 x + 5 ) + ( - 9 x ^ { 2 } + 6 )$$

Evaluate

$$11-9x-12x^{2}$$

Solution Steps

Combine -3x^{2} and -9x^{2} to get -12x^{2}.

$$-12x^{2}-9x+5+6$$

Add 5 and 6 to get 11.

$$-12x^{2}-9x+11$$

Factor

$$-12\left(x-\left(-\frac{\sqrt{609}}{24}-\frac{3}{8}\right)\right)\left(x-\left(\frac{\sqrt{609}}{24}-\frac{3}{8}\right)\right)$$

Steps Using the Quadratic Formula

Combine -3x^{2} and -9x^{2} to get -12x^{2}.

$$factor(-12x^{2}-9x+5+6)$$

Add 5 and 6 to get 11.

$$factor(-12x^{2}-9x+11)$$

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.

$$-12x^{2}-9x+11=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-12\right)\times 11}}{2\left(-12\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-12\right)\times 11}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-\left(-9\right)±\sqrt{81+48\times 11}}{2\left(-12\right)}$$

Multiply 48 times 11.

$$x=\frac{-\left(-9\right)±\sqrt{81+528}}{2\left(-12\right)}$$

Add 81 to 528.

$$x=\frac{-\left(-9\right)±\sqrt{609}}{2\left(-12\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{609}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{9±\sqrt{609}}{-24}$$

Now solve the equation x=\frac{9±\sqrt{609}}{-24} when ± is plus. Add 9 to \sqrt{609}.

$$x=\frac{\sqrt{609}+9}{-24}$$

Divide 9+\sqrt{609} by -24.

$$x=-\frac{\sqrt{609}}{24}-\frac{3}{8}$$

Now solve the equation x=\frac{9±\sqrt{609}}{-24} when ± is minus. Subtract \sqrt{609} from 9.

$$x=\frac{9-\sqrt{609}}{-24}$$

Divide 9-\sqrt{609} by -24.

$$x=\frac{\sqrt{609}}{24}-\frac{3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{8}-\frac{\sqrt{609}}{24} for x_{1} and -\frac{3}{8}+\frac{\sqrt{609}}{24} for x_{2}.

$$-12x^{2}-9x+11=-12\left(x-\left(-\frac{\sqrt{609}}{24}-\frac{3}{8}\right)\right)\left(x-\left(\frac{\sqrt{609}}{24}-\frac{3}{8}\right)\right)$$