Multiple choice

If the difference of the roots of the quadratic equation is 5 and the difference of their cubes is 215, then the quadratic equation is $x^2\,\pm\,7x\,+\,m\,=\,0$. The value of $m$ is

  1. 2

  2. 4

  3. 6

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

Let roots be a and b. Given a-b=5 and a^3-b^3=215. Since a^3-b^3 = (a-b)(a^2+ab+b^2) = (a-b)((a-b)^2 + 3ab), we have 215 = 5(25 + 3ab), so 43 = 25 + 3ab, meaning ab = 6. The equation is x^2 - (a+b)x + ab = 0. Given the form x^2 +/- 7x + m = 0, m must be 6.

AI explanation

Let the roots be alpha and beta. We are given the difference of the cubes is 215, so alpha cubed minus beta cubed equals 215. Using the identity for the difference of cubes, this becomes alpha minus beta times alpha squared plus alpha beta plus beta squared equals 215. We know alpha minus beta is 5, so alpha squared plus alpha beta plus beta squared equals 43. Squaring the difference gives alpha squared minus 2 alpha beta plus beta squared equals 25. Subtracting this from the previous equation yields 3 alpha beta equals 18, so alpha beta equals 6. In the quadratic equation x squared plus or minus 7 x plus m equals zero, the product of the roots m equals alpha beta, so m is 6.

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