Question

If \(2\) is a root of the equation \(x^2-px+q=0\) and \(p^2=4q\), then the other root is

• A. \(-2\)
• B. \(2\)
• C. \(\frac 12\)
• D. \(-\frac 12\)

Answer 1

The Correct Option is B

1. Quadratic Formula:

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

\(2 = \frac{p \pm \sqrt{p^2 - 4q}}{2}\)

2. Simplify using given information:

We know \(p^2 = 4q\):

\(2 = \frac{p \pm \sqrt{4q - 4q}}{2}\)

\(2 = \frac{p \pm 0}{2}\)

\(2 = \frac{p}{2}\)

Therefore, \(p = 4\).

3. Find the other root:

Substituting \(p=4\) into the original equation:

\(x^2 - 4x + q = 0\)

Since \(p^2 = 4q\) and \(p=4\), we have \(16 = 4q\), so \(q=4\).

The equation becomes \(x^2 - 4x + 4 = 0\)

Factoring: \((x-2)^2 = 0\)

This is a perfect square, meaning the only root is \(x=2\). It's a repeated root.