Solve (2x^2+x+1)+(-x^2-8)

$$( 2 x ^ { 2 } + x + 1 ) + ( - x ^ { 2 } - 8 )$$

Evaluate

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

Solution Steps

Subtract 8 from 1 to get -7.

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

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

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

Factor

$$\left(x-\frac{-\sqrt{29}-1}{2}\right)\left(x-\frac{\sqrt{29}-1}{2}\right)$$

Steps Using the Quadratic Formula

Subtract 8 from 1 to get -7.

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

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

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

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.

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

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.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-7\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$x=\frac{-1±\sqrt{1+28}}{2}$$

Add 1 to 28.

$$x=\frac{-1±\sqrt{29}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{29}}{2} when ± is plus. Add -1 to \sqrt{29}.

$$x=\frac{\sqrt{29}-1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{29}}{2} when ± is minus. Subtract \sqrt{29} from -1.

$$x=\frac{-\sqrt{29}-1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{29}}{2} for x_{1} and \frac{-1-\sqrt{29}}{2} for x_{2}.

$$x^{2}+x-7=\left(x-\frac{\sqrt{29}-1}{2}\right)\left(x-\frac{-\sqrt{29}-1}{2}\right)$$