Solve -5/2=-x^2-2x+3

$$- \frac{ 5 }{ 2 } =- { x }^{ 2 } -2x+3$$

Solve for x

$$x=\frac{\sqrt{26}}{2}-1\approx 1.549509757$$

$$x=-\frac{\sqrt{26}}{2}-1\approx -3.549509757$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-x^{2}-2x+3=-\frac{5}{2}$$

Add \frac{5}{2} to both sides.

$$-x^{2}-2x+3+\frac{5}{2}=0$$

Add 3 and \frac{5}{2} to get \frac{11}{2}.

$$-x^{2}-2x+\frac{11}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and \frac{11}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times \frac{11}{2}}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times \frac{11}{2}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times \frac{11}{2}}}{2\left(-1\right)}$$

Multiply 4 times \frac{11}{2}.

$$x=\frac{-\left(-2\right)±\sqrt{4+22}}{2\left(-1\right)}$$

Add 4 to 22.

$$x=\frac{-\left(-2\right)±\sqrt{26}}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±\sqrt{26}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±\sqrt{26}}{-2}$$

Now solve the equation x=\frac{2±\sqrt{26}}{-2} when ± is plus. Add 2 to \sqrt{26}.

$$x=\frac{\sqrt{26}+2}{-2}$$

Divide 2+\sqrt{26} by -2.

$$x=-\frac{\sqrt{26}}{2}-1$$

Now solve the equation x=\frac{2±\sqrt{26}}{-2} when ± is minus. Subtract \sqrt{26} from 2.

$$x=\frac{2-\sqrt{26}}{-2}$$

Divide 2-\sqrt{26} by -2.

$$x=\frac{\sqrt{26}}{2}-1$$

The equation is now solved.

$$x=-\frac{\sqrt{26}}{2}-1 x=\frac{\sqrt{26}}{2}-1$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-x^{2}-2x+3=-\frac{5}{2}$$

Subtract 3 from both sides.

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

Subtract 3 from -\frac{5}{2} to get -\frac{11}{2}.

$$-x^{2}-2x=-\frac{11}{2}$$

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=-\frac{\frac{11}{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=-\frac{\frac{11}{2}}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=-\frac{\frac{11}{2}}{-1}$$

Divide -\frac{11}{2} by -1.

$$x^{2}+2x=\frac{11}{2}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=\frac{11}{2}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{11}{2}+1$$

Add \frac{11}{2} to 1.

$$x^{2}+2x+1=\frac{13}{2}$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=\frac{13}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{13}{2}}$$

Simplify.

$$x+1=\frac{\sqrt{26}}{2} x+1=-\frac{\sqrt{26}}{2}$$

Subtract 1 from both sides of the equation.

$$x=\frac{\sqrt{26}}{2}-1 x=-\frac{\sqrt{26}}{2}-1$$