Question

If the quadratic equation \(9x^2 + 8kx + 16 = 0\) has real and equal roots, then the value of k is

• A. 3
• B. –3
• C. –4
• D. \(\pm 3\)

Hint:

When taking the square root on both sides of an algebraic equation like \(k^2 = 9\), always remember to consider both the positive and negative roots.
A common error is to only write the positive value \(k = 3\), which would lead to selecting the incorrect option (A).
Always double-check all options before finalizing your answer!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The topic of this question is Quadratic Equations.
A quadratic equation is of the standard form \(ax^2 + bx + c = 0\), where \(a \neq 0\).
The nature of the roots of a quadratic equation depends on its discriminant, denoted by \(D\).
The discriminant helps us determine whether the roots are real and distinct, real and equal, or imaginary.
For a quadratic equation to have real and equal roots, the discriminant must be exactly equal to zero.
In this problem, we are given the equation \(9x^2 + 8kx + 16 = 0\) and we need to find the value of the constant \(k\).

Step 2: Key Formula or Approach:
The discriminant \(D\) of the standard quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula:
\[ D = b^2 - 4ac \] For the equation to have real and equal roots, we set the discriminant to zero:
\[ D = 0 \implies b^2 - 4ac = 0 \] We will identify the coefficients \(a\), \(b\), and \(c\) from the given quadratic equation, substitute them into the formula, and solve the resulting equation for the variable \(k\).

Step 3: Detailed Explanation:

• Compare the given quadratic equation \(9x^2 + 8kx + 16 = 0\) with the standard quadratic equation \(ax^2 + bx + c = 0\):
From this comparison, we identify the values of the coefficients:
\(a = 9\)
\(b = 8k\)
\(c = 16\)

• Substitute these coefficients into the discriminant formula:
\[ D = (8k)^2 - 4 \cdot 9 \cdot 16 \]

• Simplify the squared term and the product of the constants:
The square of \(8k\) is \(64k^2\), and the product of \(4\), \(9\), and \(16\) is \(576\).
\[ D = 64k^2 - 576 \]

• Set the discriminant \(D\) equal to zero to satisfy the condition for real and equal roots:
\[ 64k^2 - 576 = 0 \]

• Solve for \(k^2\) by rearranging the terms of the equation:
\[ 64k^2 = 576 \] \[ k^2 = \frac{576}{64} \] \[ k^2 = 9 \]

• Take the square root on both sides of the equation to find the value of \(k\):
\[ k = \pm \sqrt{9} \] \[ k = \pm 3 \]

Step 4: Final Answer:
The values of \(k\) for which the given quadratic equation has real and equal roots are \(\pm 3\).
Therefore, the correct option is (D).