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v^2-6v=-91

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To complete the square:  we put the equation into the form ax2+bx+c = 0, or ax2+bx=-c, which is what we have with v2-6v=-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):  v2-6v+(6/2)= (-91) + (6/2)2

Simplify:  v2-6v+3= -91 + 32, or v2-6v+9 = -91 + 9 = -82

v2-6v+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 v2 - 6V = +91

Then v2-6v+32 = 91 + 32, or v2-6v+9 = 91 + 9 = 100

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