Bill F. answered • 01/06/13

Experienced Teacher & Tutor in Round Rock, TX

To complete the square: we put the equation into the form ax^{2}+bx+c = 0, or ax^{2}+bx=-c, which is what we have with v^{2}-6v=-91. the "a" in this case = 1, the "b" = -6, and c = 91. We can divide b by 2 (b/2) and add that to both sides of the equation (to keep it balanced): v^{2}-6v+(6/2)^{2 }= (-91) + (6/2)^{2}

Simplify: v^{2}-6v+3^{2 }= -91 + 3^{2}, or v^{2}-6v+9 = -91 + 9 = -82

v^{2}-6v+9 has two identical factors, v-3. Now we have (v-3)^{2} = -82

Take the square root of both sides: v-3 = √-82. But we cannot take the square root of a negative number (undefined), so we can show -82 = -(1)(+82).

V-3 = (√-1)(√82), = i√82 (√-1 = the imaginary number i)

**v = i√82+3**

Notice that the result would be much "prettier" if the original equation had been v^{2} - 6V = +91

Then v^{2}-6v+3^{2} = 91 + 3^{2}, or v^{2}-6v+9 = 91 + 9 = 100

Then we would have (v-3)^{2} = 100, or v-3 = 10, and v = 13 :-)