Solve -9x^2+6+11x^2-13-7x

$$- 9 x ^ { 2 } + 6 + 11 x ^ { 2 } - 13 - 7 x$$

Evaluate

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

Solution Steps

Combine -9x^{2} and 11x^{2} to get 2x^{2}.

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

Subtract 13 from 6 to get -7.

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

Factor

$$2\left(x-\frac{7-\sqrt{105}}{4}\right)\left(x-\frac{\sqrt{105}+7}{4}\right)$$

Steps Using the Quadratic Formula

Combine -9x^{2} and 11x^{2} to get 2x^{2}.

$$factor(2x^{2}+6-13-7x)$$

Subtract 13 from 6 to get -7.

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

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.

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

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.

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

Square -7.

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

Multiply -4 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-7\right)±\sqrt{49+56}}{2\times 2}$$

Add 49 to 56.

$$x=\frac{-\left(-7\right)±\sqrt{105}}{2\times 2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{105}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{7±\sqrt{105}}{4}$$

Now solve the equation x=\frac{7±\sqrt{105}}{4} when ± is plus. Add 7 to \sqrt{105}.

$$x=\frac{\sqrt{105}+7}{4}$$

Now solve the equation x=\frac{7±\sqrt{105}}{4} when ± is minus. Subtract \sqrt{105} from 7.

$$x=\frac{7-\sqrt{105}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{105}}{4} for x_{1} and \frac{7-\sqrt{105}}{4} for x_{2}.

$$2x^{2}-7x-7=2\left(x-\frac{\sqrt{105}+7}{4}\right)\left(x-\frac{7-\sqrt{105}}{4}\right)$$