Question

If \( -4 \) is a root of the equation \( x^2 + ax - 4 = 0 \) and the equation \( x^2 + ax + b = 0 \) has equal roots, then what will be the value of \( \sqrt{a^2 + b^2} \)?

• A. \( \frac{5}{2} \)
• B. \( \frac{5}{4} \)
• C. \( \frac{15}{4} \)
• D. \( \frac{15}{2} \)

Hint:

For equations with equal roots, use the condition \( \Delta = 0 \) (discriminant is zero) to find relations between the coefficients.

Answer 1

The Correct Option is A

For the equation \( x^2 + ax - 4 = 0 \), we substitute \( -4 \) as a root: \[ 16 - 4a - 4 = 0 \quad \Rightarrow \quad a = 3 \] For the equation \( x^2 + ax + b = 0 \) to have equal roots, the discriminant must be zero: \[ a^2 - 4b = 0 \quad \Rightarrow \quad 9 - 4b = 0 \quad \Rightarrow \quad b = \frac{9}{4} \] Now, we calculate \( \sqrt{a^2 + b^2} \): \[ \sqrt{a^2 + b^2} = \sqrt{9 + \left( \frac{9}{4} \right)^2} = \frac{5}{2} \]