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Given that a+b=12a+b=12, ab=5ab=5, find the value of 3a2+3b23a^2+3b^2

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Full solution

Q. Given that a+b=12a+b=12, ab=5ab=5, find the value of 3a2+3b23a^2+3b^2
  1. Expand and Square: We know that (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2.\newlineGiven a+b=12a+b=12, let's square it: (12)2=144(12)^2 = 144.
  2. Substitute and Simplify: Now we substitute the value of abab into the equation: 144=a2+2×5+b2144 = a^2 + 2\times5 + b^2. That simplifies to 144=a2+10+b2144 = a^2 + 10 + b^2.
  3. Rearrange Equation: We need to find 3a2+3b23a^2+3b^2, so let's rearrange the equation: 144−10=a2+b2144 - 10 = a^2 + b^2.\newlineThis gives us 134=a2+b2134 = a^2 + b^2.
  4. Multiply by 33: To find 3a2+3b23a^2+3b^2, we multiply the equation by 33: 3×134=3a2+3b23\times 134 = 3a^2 + 3b^2.\newlineSo, 3a2+3b2=4023a^2+3b^2 = 402.

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