Find the value of k if the coefficient of x^2 in the expansion of (x + k)^4 is 6

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Question: Find the value of k if the coefficient of x^2 in the expansion of (x + k)^4 is 6.

Options:

Correct Answer: 2

Solution:

The coefficient of x^2 in (x + k)^4 is C(4, 2) * k^2 = 6. Thus, 6k^2 = 6, giving k^2 = 1, so k = 1 or -1.

Find the value of k if the coefficient of x^2 in the expansion of (x + k)^4 is 6.

Find the value of k if the coefficient of x^2 in the expansion of (x + k)^4 is 6.

Step 1: Understand that we need to find the value of k in the expression (x + k)^4.
Step 2: Identify that we are looking for the coefficient of x^2 in the expansion of (x + k)^4.
Step 3: Use the binomial coefficient formula to find the coefficient of x^2, which is C(4, 2) * k^2.
Step 4: Calculate C(4, 2), which is the number of ways to choose 2 items from 4. This equals 6.
Step 5: Write the equation for the coefficient of x^2: 6 * k^2 = 6.
Step 6: Simplify the equation: Divide both sides by 6 to get k^2 = 1.
Step 7: Solve for k by taking the square root of both sides: k = 1 or k = -1.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Polynomial Expansion – Identifying and calculating specific coefficients in the expansion of polynomials.
Solving Quadratic Equations – Finding the roots of equations in the form of k^2 = c.