Solve (6x^2+8x-2)+(3x^2-9x-4)

$$(6 { x }^{ 2 } +8x-2)+(3 { x }^{ 2 } -9x-4)$$

Evaluate

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

Solution Steps

Combine 6x^{2} and 3x^{2} to get 9x^{2}.

$$9x^{2}+8x-2-9x-4$$

Combine 8x and -9x to get -x.

$$9x^{2}-x-2-4$$

Subtract 4 from -2 to get -6.

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

Factor

$$9\left(x-\frac{1-\sqrt{217}}{18}\right)\left(x-\frac{\sqrt{217}+1}{18}\right)$$

Steps Using the Quadratic Formula

Combine 6x^{2} and 3x^{2} to get 9x^{2}.

$$factor(9x^{2}+8x-2-9x-4)$$

Combine 8x and -9x to get -x.

$$factor(9x^{2}-x-2-4)$$

Subtract 4 from -2 to get -6.

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

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.

$$9x^{2}-x-6=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 9\left(-6\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-1\right)±\sqrt{1-36\left(-6\right)}}{2\times 9}$$

Multiply -36 times -6.

$$x=\frac{-\left(-1\right)±\sqrt{1+216}}{2\times 9}$$

Add 1 to 216.

$$x=\frac{-\left(-1\right)±\sqrt{217}}{2\times 9}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{217}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{1±\sqrt{217}}{18}$$

Now solve the equation x=\frac{1±\sqrt{217}}{18} when ± is plus. Add 1 to \sqrt{217}.

$$x=\frac{\sqrt{217}+1}{18}$$

Now solve the equation x=\frac{1±\sqrt{217}}{18} when ± is minus. Subtract \sqrt{217} from 1.

$$x=\frac{1-\sqrt{217}}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{217}}{18} for x_{1} and \frac{1-\sqrt{217}}{18} for x_{2}.

$$9x^{2}-x-6=9\left(x-\frac{\sqrt{217}+1}{18}\right)\left(x-\frac{1-\sqrt{217}}{18}\right)$$