Question

If \(\alpha,\beta\) are the roots of \[ ax^2+bx+c=0, \] then the quadratic equation whose roots are \(\sqrt{5}\alpha,\sqrt{5}\beta\) is

• A. \(ax^2+\sqrt{5}bx+5c=0\)
• B. \(ax^2+\sqrt{5}bx+\sqrt{5}c=0\)
• C. \(ax^2+5bx+\sqrt{5}c=0\)
• D. \(ax^2+5bx+5c=0\)

Hint:

If the roots of a quadratic equation are multiplied by a constant \(k\), then the new sum becomes \(k(\alpha+\beta)\) and the new product becomes \(k^2\alpha\beta\).

Answer 1

The Correct Option is A

Step 1: Use the substitution \(y = \sqrt{5}x\).
If \(\alpha, \beta\) are roots of \(ax^2+bx+c=0\), substitute \(x = y/\sqrt{5}\): \[a\frac{y^2}{5} + b\frac{y}{\sqrt{5}} + c = 0.\] Multiply through by 5: \[ay^2 + \sqrt{5}\,b\,y + 5c = 0.\]

Step 2: State the result.
The required quadratic (in \(y\), i.e., with roots \(\sqrt{5}\alpha, \sqrt{5}\beta\)) is \(ay^2 + \sqrt{5}\,by + 5c = 0\).
\[\boxed{ax^2 + \sqrt{5}bx + 5c = 0}\]