Question

If the equation \[ 2x^2-5x+k=0 \] has equal roots, then the value of \(k\) is

• A. \(\dfrac{25}{8}\)
• B. \(\dfrac{8}{25}\)
• C. \(\dfrac{25}{4}\)
• D. \(\dfrac{5}{2}\)

Hint:

For equal roots, immediately use \[ D=0. \] This is the fastest method and avoids unnecessary factorization.

Answer 1

The Correct Option is A

Step 1: Know the condition.
A quadratic $ax^2+bx+c=0$ has two equal (repeated) roots exactly when its discriminant is zero. The discriminant is $D=b^2-4ac$.

Step 2: Read off the coefficients.
For $2x^2-5x+k=0$, we have $a=2$, $b=-5$, and $c=k$.

Step 3: Write the equal-roots condition.
Set $D=0$: \[ b^2-4ac=0. \]

Step 4: Substitute the values.
\[ (-5)^2-4(2)(k)=0. \] \[ 25-8k=0. \]

Step 5: Solve for $k$.
\[ 8k=25 \quad\Rightarrow\quad k=\frac{25}{8}. \]

Step 6: Match the option.
The value $\dfrac{25}{8}$ is option 1.
\[ \boxed{\dfrac{25}{8}} \]