Multiple choice

If one of the roots of the equation ax2 - 11x + (2a + 2) = 0 is 8/3, and that of bx2 - 13x + (2b + 1) = 0 is 9/4 then what is the value of (b2 - a2)?

  1. 5

  2. 6

  3. 7

  4. More than one of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For ax^2 - 11x + (2a+2) = 0, root 8/3: a(64/9) - 11(8/3) + 2a + 2 = 0 => a(64/9 + 18/9) = 88/3 - 6/3 = 82/3 => a(82/9) = 82/3 => a=3. For bx^2 - 13x + (2b+1) = 0, root 9/4: b(81/16) - 13(9/4) + 2b + 1 = 0 => b(81/16 + 32/16) = 117/4 - 4/4 = 113/4 => b(113/16) = 113/4 => b=4. b^2 - a^2 = 16 - 9 = 7.

AI explanation

Substitute the root x = 8/3 into the first equation ax^2 - 11x + (2a + 2) = 0 to get a(64/9) - 88/3 + 2a + 2 = 0, which simplifies to 82a/9 = 22/3 and yields a = 3. Substitute the root x = 9/4 into the second equation bx^2 - 13x + (2b + 1) = 0 to get b(81/16) - 117/4 + 2b + 1 = 0, which simplifies to 113b/16 = 25/4 and yields b = 4. The value of (b^2 - a^2) is then 4^2 - 3^2, which equals 16 - 9, resulting in 7.

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