Question

Let \(f:\mathbb{R}\to \mathbb{R}\) be defined by \(f(x)=2x+3\). If \(\alpha,\beta\) are the roots of the equation \[ f(x^2)-2f\left(\frac{x}{2}\right)-1=0 \] then \(\alpha^2+\beta^2=\)

• A. \(13\)
• B. \(25\)
• C. \(5\)
• D. \(18\)

Hint:

When a function is given, first evaluate each required expression separately, then substitute carefully into the equation.

Answer 1

The Correct Option is C

Step 1: Substitute f into the equation.
With \(f(x) = 2x+3\), we get \(f(x^2) = 2x^2+3\) and \(f\!\left(\tfrac{x}{2}\right) = x+3\). The equation becomes \[2x^2+3 - 2(x+3) - 1 = 0 \implies 2x^2 - 2x - 4 = 0 \implies x^2 - x - 2 = 0.\]

Step 2: Find \(\alpha^2+\beta^2\).
By Vieta's formulas, \(\alpha+\beta = 1\) and \(\alpha\beta = -2\). So \[\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = 1 - 2(-2) = 5.\]
\[\boxed{\alpha^2+\beta^2 = 5}\]