Solve {r}{-2x^2+9x+5}{+x^2+3}

$$\left. \begin{array} { r } { - 2 x ^ { 2 } + 9 x + 5 } \\ { + x ^ { 2 } + 3 } \end{array} \right.$$

Evaluate

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

Solution Steps

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

$$-x^{2}+9x+5+3$$

Add 5 and 3 to get 8.

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

Factor

$$-\left(x-\frac{9-\sqrt{113}}{2}\right)\left(x-\frac{\sqrt{113}+9}{2}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-x^{2}+9x+5+3)$$

Add 5 and 3 to get 8.

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

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.

$$-x^{2}+9x+8=0$$

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.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$x=\frac{-9±\sqrt{81+32}}{2\left(-1\right)}$$

Add 81 to 32.

$$x=\frac{-9±\sqrt{113}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±\sqrt{113}}{-2}$$

Now solve the equation x=\frac{-9±\sqrt{113}}{-2} when ± is plus. Add -9 to \sqrt{113}.

$$x=\frac{\sqrt{113}-9}{-2}$$

Divide -9+\sqrt{113} by -2.

$$x=\frac{9-\sqrt{113}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{113}}{-2} when ± is minus. Subtract \sqrt{113} from -9.

$$x=\frac{-\sqrt{113}-9}{-2}$$

Divide -9-\sqrt{113} by -2.

$$x=\frac{\sqrt{113}+9}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9-\sqrt{113}}{2} for x_{1} and \frac{9+\sqrt{113}}{2} for x_{2}.

$$-x^{2}+9x+8=-\left(x-\frac{9-\sqrt{113}}{2}\right)\left(x-\frac{\sqrt{113}+9}{2}\right)$$