Solve (x)=-x^2+4x+6

$$( x ) = - x ^ { 2 } + 4 x + 6$$

Solve for x

$$x = \frac{\sqrt{33} + 3}{2} \approx 4.372281323$$

$$x=\frac{3-\sqrt{33}}{2}\approx -1.372281323$$

Steps Using the Quadratic Formula

Add x^{2} to both sides.

$$x+x^{2}=4x+6$$

Subtract 4x from both sides.

$$x+x^{2}-4x=6$$

Combine x and -4x to get -3x.

$$-3x+x^{2}=6$$

Subtract 6 from both sides.

$$-3x+x^{2}-6=0$$

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.

$$x^{2}-3x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-6\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-\left(-3\right)±\sqrt{9+24}}{2}$$

Add 9 to 24.

$$x=\frac{-\left(-3\right)±\sqrt{33}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{33}}{2}$$

Now solve the equation x=\frac{3±\sqrt{33}}{2} when ± is plus. Add 3 to \sqrt{33}.

$$x=\frac{\sqrt{33}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{33}}{2} when ± is minus. Subtract \sqrt{33} from 3.

$$x=\frac{3-\sqrt{33}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{33}+3}{2} x=\frac{3-\sqrt{33}}{2}$$

Steps for Completing the Square

Add x^{2} to both sides.

$$x+x^{2}=4x+6$$

Subtract 4x from both sides.

$$x+x^{2}-4x=6$$

Combine x and -4x to get -3x.

$$-3x+x^{2}=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-3x=6$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=6+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=6+\frac{9}{4}$$

Add 6 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{33}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{33}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{33}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{33}}{2} x-\frac{3}{2}=-\frac{\sqrt{33}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{33}+3}{2} x=\frac{3-\sqrt{33}}{2}$$