Solve -6x^2-7+2x+8x^2-13x

$$- 6 x ^ { 2 } - 7 + 2 x + 8 x ^ { 2 } - 13 x$$

Evaluate

$$2x^{2}-11x-7$$

Solution Steps

Combine -6x^{2} and 8x^{2} to get 2x^{2}.

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

Combine 2x and -13x to get -11x.

$$2x^{2}-7-11x$$

Factor

$$2\left(x-\frac{11-\sqrt{177}}{4}\right)\left(x-\frac{\sqrt{177}+11}{4}\right)$$

Steps Using the Quadratic Formula

Combine -6x^{2} and 8x^{2} to get 2x^{2}.

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

Combine 2x and -13x to get -11x.

$$factor(2x^{2}-7-11x)$$

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.

$$2x^{2}-11x-7=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-11\right)±\sqrt{121-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-11\right)±\sqrt{121+56}}{2\times 2}$$

Add 121 to 56.

$$x=\frac{-\left(-11\right)±\sqrt{177}}{2\times 2}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{177}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{11±\sqrt{177}}{4}$$

Now solve the equation x=\frac{11±\sqrt{177}}{4} when ± is plus. Add 11 to \sqrt{177}.

$$x=\frac{\sqrt{177}+11}{4}$$

Now solve the equation x=\frac{11±\sqrt{177}}{4} when ± is minus. Subtract \sqrt{177} from 11.

$$x=\frac{11-\sqrt{177}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{177}}{4} for x_{1} and \frac{11-\sqrt{177}}{4} for x_{2}.

$$2x^{2}-11x-7=2\left(x-\frac{\sqrt{177}+11}{4}\right)\left(x-\frac{11-\sqrt{177}}{4}\right)$$