Multiple choice

If α and β are the roots of the quadratic equation x2−10x+15=0, then find the quadratic equation whose roots are α + α/β] and [β + β/α]

  1. 15x2+71x+210= 0

  2. 5x2−22x+56= 0

  3. 3x2−44x+78= 0

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The roots satisfy α + β = 10 and αβ = 15. The transformed roots have sum α + β + α/β + β/α = 10 + (100 - 30)/15 = 44/3, and product (α + 1)(β + 1) = 26. Hence their equation is x^2 - (44/3)x + 26 = 0, or 3x^2 - 44x + 78 = 0.

AI explanation

Using Vieta's formulas for x2 minus 10x plus 15 equals 0, the sum of the roots alpha plus beta is 10 and the product alpha times beta is 15. The new roots simplify to alpha times (beta plus 1) over beta and beta times (alpha plus 1) over alpha, making their sum equal to (alpha squared times beta plus alpha squared plus beta squared times alpha plus beta squared) divided by alpha times beta. Substituting the values gives a sum of 44 over 3, and the product simplifies to alpha plus beta plus 2, which equals 12. The new quadratic equation is x squared minus (sum over product)x plus 1 equals 0, yielding 3x squared minus 44x plus 78 equals 0.

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