Multiple choice

The quadratic equations $\displaystyle x^{2}-6x+a= 0$ and $\displaystyle x^{2}-cx+6= 0$ have one root in common. The other roots of the first & the second equations are integers in the ratio $4 : 3$. then the common root is

  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let common root be k. k^2 - 6k + a = 0 and k^2 - ck + 6 = 0. Other roots are r1 = 6/k and r2 = a/k. Ratio r1/r2 = 4/3 or 3/4. If k=2, r1=3, r2=a/2. If r1/r2 = 3/4, 3/(a/2) = 3/4 => a/2=4 => a=8. Check: x^2-6x+8=(x-2)(x-4), x^2-cx+6=(x-2)(x-3). Common root is 2.

AI explanation

Let the roots of the first equation x^2 - 6x + a = 0 be k and 4y, and the roots of the second equation x^2 - cx + 6 = 0 be k and 3y, where k is the common root. Using the product of roots for the second equation, we have k(3y) = 6, so ky = 2. Using the product of roots for the first equation, we have k(4y) = a, so 4y = 2a/k. Since the sum of the roots of the first equation is 6, we get k + 4y = 6. Testing the integer possibilities from ky = 2, if k = 2, then y = 1, which perfectly satisfies k + 4y = 6 since 2 + 4(1) = 6. Therefore, the common root is 2.

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