Solve (12x-6)+(6x-2+x^2)

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

Evaluate

$$x^{2}+18x-8$$

Solution Steps

Combine 12x and 6x to get 18x.

$$18x-6-2+x^{2}$$

Subtract 2 from -6 to get -8.

$$18x-8+x^{2}$$

Factor

$$\left(x-\left(-\sqrt{89}-9\right)\right)\left(x-\left(\sqrt{89}-9\right)\right)$$

Steps Using the Quadratic Formula

Combine 12x and 6x to get 18x.

$$factor(18x-6-2+x^{2})$$

Subtract 2 from -6 to get -8.

$$factor(18x-8+x^{2})$$

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.

$$x^{2}+18x-8=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-8\right)}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$x=\frac{-18±\sqrt{324+32}}{2}$$

Add 324 to 32.

$$x=\frac{-18±\sqrt{356}}{2}$$

Take the square root of 356.

$$x=\frac{-18±2\sqrt{89}}{2}$$

Now solve the equation x=\frac{-18±2\sqrt{89}}{2} when ± is plus. Add -18 to 2\sqrt{89}.

$$x=\frac{2\sqrt{89}-18}{2}$$

Divide -18+2\sqrt{89} by 2.

$$x=\sqrt{89}-9$$

Now solve the equation x=\frac{-18±2\sqrt{89}}{2} when ± is minus. Subtract 2\sqrt{89} from -18.

$$x=\frac{-2\sqrt{89}-18}{2}$$

Divide -18-2\sqrt{89} by 2.

$$x=-\sqrt{89}-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -9+\sqrt{89} for x_{1} and -9-\sqrt{89} for x_{2}.

$$x^{2}+18x-8=\left(x-\left(\sqrt{89}-9\right)\right)\left(x-\left(-\sqrt{89}-9\right)\right)$$