Solve -7x^2-6+9x+8x^2-12x

$$- 7 x ^ { 2 } - 6 + 9 x + 8 x ^ { 2 } - 12 x$$

Evaluate

$$x^{2}-3x-6$$

Solution Steps

Combine -7x^{2} and 8x^{2} to get x^{2}.

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

Combine 9x and -12x to get -3x.

$$x^{2}-6-3x$$

Factor

$$\left(x-\frac{3-\sqrt{33}}{2}\right)\left(x-\frac{\sqrt{33}+3}{2}\right)$$

Steps Using the Quadratic Formula

Combine -7x^{2} and 8x^{2} to get x^{2}.

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

Combine 9x and -12x to get -3x.

$$factor(x^{2}-6-3x)$$

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.

$$x^{2}-3x-6=0$$

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.

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

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-\left(-3\right)±\sqrt{9+24}}{2}$$

Add 9 to 24.

$$x=\frac{-\left(-3\right)±\sqrt{33}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{33}}{2}$$

Now solve the equation x=\frac{3±\sqrt{33}}{2} when ± is plus. Add 3 to \sqrt{33}.

$$x=\frac{\sqrt{33}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{33}}{2} when ± is minus. Subtract \sqrt{33} from 3.

$$x=\frac{3-\sqrt{33}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{33}}{2} for x_{1} and \frac{3-\sqrt{33}}{2} for x_{2}.

$$x^{2}-3x-6=\left(x-\frac{\sqrt{33}+3}{2}\right)\left(x-\frac{3-\sqrt{33}}{2}\right)$$