Multiple choice

What is the minimum value of the sum of the squares of the roots of the equation 𝑥²﹣(𝑝﹣3)𝑥 + (𝑝﹣7) = 0?

  1. 5

  2. 7

  3. 18

  4. 10
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sum of squares = (alpha + beta)^2 - 2(alpha * beta) = (p-3)^2 - 2(p-7) = p^2 - 6p + 9 - 2p + 14 = p^2 - 8p + 23. Minimum value occurs at p = -b/2a = 8/2 = 4. Min value = 16 - 32 + 23 = 7.

AI explanation

Let the roots be r and s. Using Vieta's formulas, the sum of the roots r + s equals p - 3, and the product rs equals p - 7. The sum of the squares of the roots is r squared plus s squared, which equals the quantity r + s squared minus 2rs. Substituting the known values gives the expression p minus 3 squared minus 2 times the quantity p minus 7, which simplifies to p squared minus 8p plus 23. To find the minimum value of this quadratic expression, we calculate the value at p = 4, resulting in 16 minus 32 plus 23, which equals 7.

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