Question

One of the roots of the equation x² + kx - 12 = 0 is 3, and k is a constant.
Quantity A: The value of k
Quantity B: -1

• A. The two quantities are equal.
• B. Quantity A is greater.
• C. Quantity B is greater.
• D. The relationship cannot be determined from the information given.
• E.

Hint:

This problem tests the fundamental definition of a root. Remember that if 'r' is a root of a polynomial P(x), then P(r) = 0. This is a very common type of question in algebra.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A root of an equation is a value that, when substituted for the variable, makes the equation true. If we know a root of a polynomial equation, we can use it to find the value of any unknown coefficients.
Step 2: Key Formula or Approach:
Substitute the given root, x = 3, into the equation x² + kx - 12 = 0 and solve for the constant k.
Step 3: Detailed Explanation:
The given equation is: \[ x^2 + kx - 12 = 0 \] We are told that x = 3 is a root. Substitute this value into the equation: \[ (3)^2 + k(3) - 12 = 0 \] \[ 9 + 3k - 12 = 0 \] Combine the constant terms: \[ 3k - 3 = 0 \] Add 3 to both sides: \[ 3k = 3 \] Divide by 3: \[ k = 1 \] So, Quantity A, the value of k, is 1.
Quantity B is -1.
Now we compare Quantity A and Quantity B.
\[ 1>-1 \] Step 4: Final Answer:
Quantity A is greater than Quantity B.