Question

If the roots of the quadratic equation \(5x^2 - 10x + k = 0\) are real and equal, then the value of k is :

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

For a quadratic equation \(ax^2 + bx + c = 0\) to have equal roots, it must be a perfect square trinomial.
Rewriting: \(5(x^2 - 2x) + k = 0 \implies 5(x^2 - 2x + 1) = 0 \implies 5x^2 - 10x + 5 = 0\).
By comparing this directly with the given equation, we immediately see that \(k = 5\).

Answer 1

The Correct Option is B

Step 1: Rewrite the equation by completing the square.
Take $5x^2 - 10x + k = 0$ and pull 5 out of the first two terms: \[ 5(x^2 - 2x) + k = 0 \] \[ 5[(x-1)^2 - 1] + k = 0 \] \[ 5(x-1)^2 + (k - 5) = 0 \]
Step 2: Think about when this gives equal roots.
A perfect square term like $5(x-1)^2$ alone would give a double root at $x=1$. For that to actually be the full picture, the leftover constant $(k-5)$ must vanish, otherwise we would not get a clean repeated root.
Step 3: Solve and conclude. \[ k - 5 = 0 \implies k = 5 \] \[ \boxed{k = 5} \]