Solve (5x^2-3x+9)+(-11x^2-13x-16)

$$( 5 x ^ { 2 } - 3 x + 9 ) + ( - 11 x ^ { 2 } - 13 x - 16 )$$

Evaluate

$$-6x^{2}-16x-7$$

Solution Steps

Combine 5x^{2} and -11x^{2} to get -6x^{2}.

$$-6x^{2}-3x+9-13x-16$$

Combine -3x and -13x to get -16x.

$$-6x^{2}-16x+9-16$$

Subtract 16 from 9 to get -7.

$$-6x^{2}-16x-7$$

Factor

$$-6\left(x-\left(-\frac{\sqrt{22}}{6}-\frac{4}{3}\right)\right)\left(x-\left(\frac{\sqrt{22}}{6}-\frac{4}{3}\right)\right)$$

Steps Using the Quadratic Formula

Combine 5x^{2} and -11x^{2} to get -6x^{2}.

$$factor(-6x^{2}-3x+9-13x-16)$$

Combine -3x and -13x to get -16x.

$$factor(-6x^{2}-16x+9-16)$$

Subtract 16 from 9 to get -7.

$$factor(-6x^{2}-16x-7)$$

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.

$$-6x^{2}-16x-7=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-6\right)\left(-7\right)}}{2\left(-6\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-6\right)\left(-7\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-16\right)±\sqrt{256+24\left(-7\right)}}{2\left(-6\right)}$$

Multiply 24 times -7.

$$x=\frac{-\left(-16\right)±\sqrt{256-168}}{2\left(-6\right)}$$

Add 256 to -168.

$$x=\frac{-\left(-16\right)±\sqrt{88}}{2\left(-6\right)}$$

Take the square root of 88.

$$x=\frac{-\left(-16\right)±2\sqrt{22}}{2\left(-6\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{22}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{16±2\sqrt{22}}{-12}$$

Now solve the equation x=\frac{16±2\sqrt{22}}{-12} when ± is plus. Add 16 to 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+16}{-12}$$

Divide 16+2\sqrt{22} by -12.

$$x=-\frac{\sqrt{22}}{6}-\frac{4}{3}$$

Now solve the equation x=\frac{16±2\sqrt{22}}{-12} when ± is minus. Subtract 2\sqrt{22} from 16.

$$x=\frac{16-2\sqrt{22}}{-12}$$

Divide 16-2\sqrt{22} by -12.

$$x=\frac{\sqrt{22}}{6}-\frac{4}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{4}{3}-\frac{\sqrt{22}}{6} for x_{1} and -\frac{4}{3}+\frac{\sqrt{22}}{6} for x_{2}.

$$-6x^{2}-16x-7=-6\left(x-\left(-\frac{\sqrt{22}}{6}-\frac{4}{3}\right)\right)\left(x-\left(\frac{\sqrt{22}}{6}-\frac{4}{3}\right)\right)$$