Solve -x^2+18x+27

$$- { x }^{ 2 } +18x+27$$

Factor

$$-\left(x-\left(9-6\sqrt{3}\right)\right)\left(x-\left(6\sqrt{3}+9\right)\right)$$

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.

$$-x^{2}+18x+27=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-1\right)\times 27}}{2\left(-1\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-1\right)\times 27}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-18±\sqrt{324+4\times 27}}{2\left(-1\right)}$$

Multiply 4 times 27.

$$x=\frac{-18±\sqrt{324+108}}{2\left(-1\right)}$$

Add 324 to 108.

$$x=\frac{-18±\sqrt{432}}{2\left(-1\right)}$$

Take the square root of 432.

$$x=\frac{-18±12\sqrt{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-18±12\sqrt{3}}{-2}$$

Now solve the equation x=\frac{-18±12\sqrt{3}}{-2} when ± is plus. Add -18 to 12\sqrt{3}.

$$x=\frac{12\sqrt{3}-18}{-2}$$

Divide -18+12\sqrt{3} by -2.

$$x=9-6\sqrt{3}$$

Now solve the equation x=\frac{-18±12\sqrt{3}}{-2} when ± is minus. Subtract 12\sqrt{3} from -18.

$$x=\frac{-12\sqrt{3}-18}{-2}$$

Divide -18-12\sqrt{3} by -2.

$$x=6\sqrt{3}+9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9-6\sqrt{3} for x_{1} and 9+6\sqrt{3} for x_{2}.

$$-x^{2}+18x+27=-\left(x-\left(9-6\sqrt{3}\right)\right)\left(x-\left(6\sqrt{3}+9\right)\right)$$

Evaluate

$$27+18x-x^{2}$$