Solve -x^2+3x+9

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

Factor

$$-\left(x-\frac{3-3\sqrt{5}}{2}\right)\left(x-\frac{3\sqrt{5}+3}{2}\right)$$

Steps Using the Quadratic Formula

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.

$$-x^{2}+3x+9=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\times 9}}{2\left(-1\right)}$$

Multiply 4 times 9.

$$x=\frac{-3±\sqrt{9+36}}{2\left(-1\right)}$$

Add 9 to 36.

$$x=\frac{-3±\sqrt{45}}{2\left(-1\right)}$$

Take the square root of 45.

$$x=\frac{-3±3\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±3\sqrt{5}}{-2}$$

Now solve the equation x=\frac{-3±3\sqrt{5}}{-2} when ± is plus. Add -3 to 3\sqrt{5}.

$$x=\frac{3\sqrt{5}-3}{-2}$$

Divide -3+3\sqrt{5} by -2.

$$x=\frac{3-3\sqrt{5}}{2}$$

Now solve the equation x=\frac{-3±3\sqrt{5}}{-2} when ± is minus. Subtract 3\sqrt{5} from -3.

$$x=\frac{-3\sqrt{5}-3}{-2}$$

Divide -3-3\sqrt{5} by -2.

$$x=\frac{3\sqrt{5}+3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-3\sqrt{5}}{2} for x_{1} and \frac{3+3\sqrt{5}}{2} for x_{2}.

$$-x^{2}+3x+9=-\left(x-\frac{3-3\sqrt{5}}{2}\right)\left(x-\frac{3\sqrt{5}+3}{2}\right)$$

Evaluate

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