Q5 - Completing the square with multiple quadratic terms


Question 5: $$ \text{If } a^2 + 5b^2 + 4c^2 - 4ab - 4bc = 0 \text{ then find the value of } \frac{a^2 + ac}{b^2}. $$ Solution:
1. First we will complete the square in "a" since a^2 has the coefficient of 1.
So we get:
\((a -2b)^2 + ... = 0\)
2. Now we complete the square of the remaining terms to get:
\((a - 2b)^2 + (b - 2c)^2 = 0.\)
If two squares add up to 0 then they are individually 0.
So a = 2b and b = 2c.
Put the value of a,c in terms of b in the question to get the answer which is 5.

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