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# Solve a quadratic equation by completing the square

With quadratic equations (ax2 + bx + c = 0), you can solve by completing the square.

Step 1: Make sure that the left side of the equation looks like x2 + bx.

Step 2: Add $(\frac{b}{2})^2$ to both sides.

Step 3: Factor the left side as $(x+\frac{b}{2})^2$

Step 4: Take the square root and solve.

Solve by completing the square.

x2 - 2x - 3 = 0
Step 1: Make sure that the left side of the equation looks like x2 + bx.

To make the left side of the equation look like x2 + bx, add 3 to both sides.

 x2 - 2x - 3 = 0 x2 - 2x = 3

Step 2: Add $(\frac{b}{2})^2$ to both sides.

Since b=-2, $(\frac{b}{2})^2\;=\;(\frac{-2}{2})^2\;=\;(-1)^2\;=\;1$

 x2 - 2x = 3 x2 - 2x + 1 = 4

Step 3: Factor the left side as $(x+\frac{b}{2})^2$

In general, an expression of the form $(x^2+bx+\frac{b}{2})^2$ can be factored as $(x+\frac{b}{2})^2$

The expression x2 - 2x + 1 is of this form, with b=-2. So, it can be factored as (x-1)2.

Rewrite the equation with the left side factored.

(x-1)2 = 4

Step 4: Take the square root and solve.

 x - 1 = $\pm\;2$ Take the square root x = $1\;\pm\;2$ Add 1 to both sides x = 1+2   or   x = 1-2         Split $\pm$ into + or - x=3 or x=-1 Simplify

Solve for by completing the square

t2-13t+36 = 0
t = or t =