Quadratic Equations — Questions & Answers

Sample questions from Quadratic Equations

1. Q1. A quadratic equation can have at most how many roots?

   • A.2✓
   • B.1
   • C.3
   • D.4
   Solution

   A degree-two equation has at most two roots.

2. Q2. If x = 3 is a solution of x^2 - 2kx - 6 = 0, then k equals:

   • A.1/2✓
   • B.1
   • C.3/2
   • D.2
   Solution

   Substituting x=3: 9 - 6k - 6 = 0, so 3 - 6k = 0, giving k = 1/2.

3. Q3. The equation x^2 - (2+m)x + (m-2)^2 = 0 has equal roots for:

   • A.m = 2 or m = 6
   • B.m = 2/3 or m = 6✓
   • C.m = -2 or m = 3
   • D.m = 4 only
   Solution

   D = (2+m)^2 - 4(m-2)^2 = 0. Expanding: (m^2+4m+4) - 4(m^2-4m+4) = -3m^2+20m-12 = 0, i.e. 3m^2-20m+12=0. D' = 400-144=256=16^2, so m = (20±16)/6 = 6 or 2/3.

4. Q4. The roots of x²−16=0 are:

   • A.4,−4✓
   • B.16,−16
   • C.8,−8
   • D.2,−2
   Solution

   x²=16 gives x=±4.

5. Q5. If alpha and beta are the roots of 3x^2 - 4x - 4 = 0, the value of 1/alpha + 1/beta is:

   • A.-1✓
   • B.1
   • C.3
   • D.-3
   Solution

   Sum alpha+beta = 4/3, product alpha*beta = -4/3. So 1/alpha + 1/beta = (alpha+beta)/(alpha*beta) = (4/3)/(-4/3) = -1.

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