Question

If the equation \( 2x^2 + (a+3)x + 8 = 0 \) has equal roots, then one of the values of \( a \) is:

• A. \(-9 \)
• B. \(-5 \)
• C. \(-11 \)
• D. \( 11 \)
• E. \( 9 \)

Hint:

When a question asks for "one of the values," always be prepared for the square root to yield both a positive and a negative result. If your first calculated value isn't in the options, check the alternative sign from the square root.

Answer 1

The Correct Option is C

Concept: For a quadratic equation of the form \( Ax^2 + Bx + C = 0 \), the condition for having equal roots (also known as repeated roots or a discriminant of zero) is: \[ D = B^2 - 4AC = 0 \] This condition arises because the quadratic formula \( x = \frac{-B \pm \sqrt{D}}{2A} \) yields only one distinct value when the term under the square root vanishes.

Step 1: Identify the coefficients from the given equation.
Comparing \( 2x^2 + (a+3)x + 8 = 0 \) with the standard form \( Ax^2 + Bx + C = 0 \), we identify:
• \( A = 2 \)
• \( B = (a+3) \)
• \( C = 8 \)

Step 2: Apply the condition for equal roots (\( D = 0 \)).
Substitute the identified coefficients into the discriminant formula: \[ (a+3)^2 - 4(2)(8) = 0 \] \[ (a+3)^2 - 64 = 0 \]

Step 3: Solve the resulting equation for \( a \).
Rearrange the equation to isolate the squared term: \[ (a+3)^2 = 64 \] Taking the square root on both sides provides two possible linear equations: \[ a+3 = 8 \quad \text{or} \quad a+3 = -8 \] Solving for \( a \) in both cases: \[ \text{Case 1: } a = 8 - 3 = 5 \] \[ \text{Case 2: } a = -8 - 3 = -11 \]