Multiple choice

The values of a and b so that x4 + 12x3 + 46x2 + ax + b is square of a quadratic expression are respectively?

  1. 60, 25

  2. 45, 25

  3. 60, 30

  4. 25, 45
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A Correct answer
Explanation

For x^4 + 12x^3 + 46x^2 + ax + b to be a perfect square (x^2 + 6x + k)^2 = x^4 + 12x^3 + (36 + 2k)x^2 + 12kx + k^2. Matching coefficients: 36 + 2k = 46 => 2k = 10 => k = 5. Then a = 12k = 60 and b = k^2 = 25.

AI explanation

Assume the given expression is the square of the quadratic (x^2 + bx + c). Expanding this square gives x^4 + 2bx^3 + (b^2 + 2c)x^2 + 2bcx + c^2. Matching coefficients with x^4 + 12x^3 + 46x^2 + ax + b gives 2b = 12, so b = 6. Using the x^2 coefficient, 36 + 2c = 46, so c = 5. Therefore, a = 2(6)(5) = 60 and b = 25.

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