Multiple choice

If a and b are the roots of a quadratic equation such that ${a^2}b + {b^2}a = 84,$ also a and b are natural numbers, then equation can be

  1. ${x^2} + 21x - 4 = 0$
  2. ${x^2} - 21x + 4 = 0$
  3. ${x^2} - 7x + 12 = 0$
  4. ${x^2} - 12x + 7 = 0$
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C Correct answer
Explanation

a^2b + b^2a = ab(a+b) = 84. Since a, b are natural numbers, we test factors of 84. If a=3, b=4: 3*4*(3+4) = 12*7 = 84. The quadratic equation with roots 3 and 4 is (x-3)(x-4) = x^2 - 7x + 12 = 0.

AI explanation

Factor the given expression a^2b + ab^2 = 84 by factoring out the common term ab, giving ab(a + b) = 84. In a quadratic equation where a and b are the roots, the sum (a + b) and the product ab must both be integers because a and b are natural numbers. By testing integer factors of 84, we find that if the sum is 7 and the product is 12, the equation ab(a + b) equals 12 * 7, which is exactly 84. Using Vieta's formulas for a quadratic equation x^2 - (sum)x + (product) = 0, we substitute the sum of 7 and the product of 12. The resulting quadratic equation is x^2 - 7x + 12 = 0.

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