Solve 2.5=x^2-2x-3

$$2.5 = x ^ { 2 } - 2 x - 3$$

Solve for x

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

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

Steps Using the Quadratic Formula

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

$$x^{2}-2x-3=2.5$$

Subtract 2.5 from both sides.

$$x^{2}-2x-3-2.5=0$$

Subtract 2.5 from -3 to get -5.5.

$$x^{2}-2x-5.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -5.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-5.5\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-5.5\right)}}{2}$$

Multiply -4 times -5.5.

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

Add 4 to 22.

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

The opposite of -2 is 2.

$$x=\frac{2±\sqrt{26}}{2}$$

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.

$$x^{2}-2x-3=2.5$$

Add 3 to both sides.

$$x^{2}-2x=2.5+3$$

Add 2.5 and 3 to get 5.5.

$$x^{2}-2x=5.5$$

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.

$$x^{2}-2x+1=5.5+1$$

Add 5.5 to 1.

$$x^{2}-2x+1=6.5$$

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}.

$$\left(x-1\right)^{2}=6.5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{6.5}$$

Simplify.

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

Add 1 to both sides of the equation.

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