Solve -x^2+2x+3=-(x+9/4)^2+2xx+9/4)+3

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

Solve for x

$$x = \frac{\sqrt{355} + 13}{8} \approx 3.98018046$$

$$x=\frac{13-\sqrt{355}}{8}\approx -0.73018046$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$-x^{2}+2x+3=-\left(x+\frac{9}{4}\right)^{2}+2x^{2}+\frac{9}{4}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\frac{9}{4}\right)^{2}.

$$-x^{2}+2x+3=-\left(x^{2}+\frac{9}{2}x+\frac{81}{16}\right)+2x^{2}+\frac{9}{4}$$

To find the opposite of x^{2}+\frac{9}{2}x+\frac{81}{16}, find the opposite of each term.

$$-x^{2}+2x+3=-x^{2}-\frac{9}{2}x-\frac{81}{16}+2x^{2}+\frac{9}{4}$$

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

$$-x^{2}+2x+3=x^{2}-\frac{9}{2}x-\frac{81}{16}+\frac{9}{4}$$

Add -\frac{81}{16} and \frac{9}{4} to get -\frac{45}{16}.

$$-x^{2}+2x+3=x^{2}-\frac{9}{2}x-\frac{45}{16}$$

Subtract x^{2} from both sides.

$$-x^{2}+2x+3-x^{2}=-\frac{9}{2}x-\frac{45}{16}$$

Add \frac{9}{2}x to both sides.

$$-x^{2}+2x+3-x^{2}+\frac{9}{2}x=-\frac{45}{16}$$

Combine 2x and \frac{9}{2}x to get \frac{13}{2}x.

$$-x^{2}+\frac{13}{2}x+3-x^{2}=-\frac{45}{16}$$

Add \frac{45}{16} to both sides.

$$-x^{2}+\frac{13}{2}x+3-x^{2}+\frac{45}{16}=0$$

Add 3 and \frac{45}{16} to get \frac{93}{16}.

$$-x^{2}+\frac{13}{2}x+\frac{93}{16}-x^{2}=0$$

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

$$-2x^{2}+\frac{13}{2}x+\frac{93}{16}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, \frac{13}{2} for b, and \frac{93}{16} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{13}{2}±\sqrt{\left(\frac{13}{2}\right)^{2}-4\left(-2\right)\times \frac{93}{16}}}{2\left(-2\right)}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{13}{2}±\sqrt{\frac{169}{4}-4\left(-2\right)\times \frac{93}{16}}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\frac{13}{2}±\sqrt{\frac{169}{4}+8\times \frac{93}{16}}}{2\left(-2\right)}$$

Multiply 8 times \frac{93}{16}.

$$x=\frac{-\frac{13}{2}±\sqrt{\frac{169}{4}+\frac{93}{2}}}{2\left(-2\right)}$$

Add \frac{169}{4} to \frac{93}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{13}{2}±\sqrt{\frac{355}{4}}}{2\left(-2\right)}$$

Take the square root of \frac{355}{4}.

$$x=\frac{-\frac{13}{2}±\frac{\sqrt{355}}{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-\frac{13}{2}±\frac{\sqrt{355}}{2}}{-4}$$

Now solve the equation x=\frac{-\frac{13}{2}±\frac{\sqrt{355}}{2}}{-4} when ± is plus. Add -\frac{13}{2} to \frac{\sqrt{355}}{2}.

$$x=\frac{\sqrt{355}-13}{-4\times 2}$$

Divide \frac{-13+\sqrt{355}}{2} by -4.

$$x=\frac{13-\sqrt{355}}{8}$$

Now solve the equation x=\frac{-\frac{13}{2}±\frac{\sqrt{355}}{2}}{-4} when ± is minus. Subtract \frac{\sqrt{355}}{2} from -\frac{13}{2}.

$$x=\frac{-\sqrt{355}-13}{-4\times 2}$$

Divide \frac{-13-\sqrt{355}}{2} by -4.

$$x=\frac{\sqrt{355}+13}{8}$$

The equation is now solved.

$$x=\frac{13-\sqrt{355}}{8} x=\frac{\sqrt{355}+13}{8}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$-x^{2}+2x+3=-\left(x+\frac{9}{4}\right)^{2}+2x^{2}+\frac{9}{4}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\frac{9}{4}\right)^{2}.

$$-x^{2}+2x+3=-\left(x^{2}+\frac{9}{2}x+\frac{81}{16}\right)+2x^{2}+\frac{9}{4}$$

To find the opposite of x^{2}+\frac{9}{2}x+\frac{81}{16}, find the opposite of each term.

$$-x^{2}+2x+3=-x^{2}-\frac{9}{2}x-\frac{81}{16}+2x^{2}+\frac{9}{4}$$

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

$$-x^{2}+2x+3=x^{2}-\frac{9}{2}x-\frac{81}{16}+\frac{9}{4}$$

Add -\frac{81}{16} and \frac{9}{4} to get -\frac{45}{16}.

$$-x^{2}+2x+3=x^{2}-\frac{9}{2}x-\frac{45}{16}$$

Subtract x^{2} from both sides.

$$-x^{2}+2x+3-x^{2}=-\frac{9}{2}x-\frac{45}{16}$$

Add \frac{9}{2}x to both sides.

$$-x^{2}+2x+3-x^{2}+\frac{9}{2}x=-\frac{45}{16}$$

Combine 2x and \frac{9}{2}x to get \frac{13}{2}x.

$$-x^{2}+\frac{13}{2}x+3-x^{2}=-\frac{45}{16}$$

Subtract 3 from both sides.

$$-x^{2}+\frac{13}{2}x-x^{2}=-\frac{45}{16}-3$$

Subtract 3 from -\frac{45}{16} to get -\frac{93}{16}.

$$-x^{2}+\frac{13}{2}x-x^{2}=-\frac{93}{16}$$

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

$$-2x^{2}+\frac{13}{2}x=-\frac{93}{16}$$

Divide both sides by -2.

$$\frac{-2x^{2}+\frac{13}{2}x}{-2}=-\frac{\frac{93}{16}}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{\frac{13}{2}}{-2}x=-\frac{\frac{93}{16}}{-2}$$

Divide \frac{13}{2} by -2.

$$x^{2}-\frac{13}{4}x=-\frac{\frac{93}{16}}{-2}$$

Divide -\frac{93}{16} by -2.

$$x^{2}-\frac{13}{4}x=\frac{93}{32}$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=\frac{93}{32}+\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{93}{32}+\frac{169}{64}$$

Add \frac{93}{32} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{355}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{355}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{355}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{\sqrt{355}}{8} x-\frac{13}{8}=-\frac{\sqrt{355}}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=\frac{\sqrt{355}+13}{8} x=\frac{13-\sqrt{355}}{8}$$