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 \(\alpha, \beta\) are roots of \(ax^2+bx+c=0\), then \(\alpha+\beta = -b/a\) and \(\alpha\beta = c/a\).
To form a new quadratic equation with roots \(k\alpha, k\beta\): Let the new variable be \(y = kx\), so \(x = y/k\). Substitute this into the original equation \(a(y/k)^2 + b(y/k) + c = 0\). \(a y^2/k^2 + by/k + c = 0\). Multiply by \(k^2\): \(ay^2 + bky + ck^2 = 0\). Replace y with x: \(ax^2 + bkx + ck^2 = 0\).
In this problem, \(k=\sqrt{5}\). So, \(k^2=5\). The new equation is \(ax^2 + b(\sqrt{5})x + c(\sqrt{5})^2 = 0 \Rightarrow ax^2 + \sqrt{5}bx + 5c = 0\).

Answer 1

The Correct Option is A

Given that \(\alpha, \beta\) are the roots of \(ax^2+bx+c=0\). From Vieta's formulas: Sum of roots: \(\alpha + \beta = -b/a\) Product of roots: \(\alpha\beta = c/a\) We need to form a new quadratic equation whose roots are \(\alpha' = \sqrt{5}\alpha\) and \(\beta' = \sqrt{5}\beta\). Let the new equation be \(Ax^2+Bx+C=0\). Sum of new roots: \(\alpha' + \beta' = \sqrt{5}\alpha + \sqrt{5}\beta = \sqrt{5}(\alpha+\beta)\). Substitute \(\alpha+\beta = -b/a\): \(\alpha' + \beta' = \sqrt{5}(-b/a) = -\frac{\sqrt{5}b}{a}\). This is equal to \(-B/A\) for the new equation. Product of new roots: \(\alpha' \beta' = (\sqrt{5}\alpha)(\sqrt{5}\beta) = 5\alpha\beta\). Substitute \(\alpha\beta = c/a\): \(\alpha' \beta' = 5(c/a) = \frac{5c}{a}\). This is equal to \(C/A\) for the new equation. A quadratic equation with roots \(\alpha', \beta'\) can be written as \(k[x^2 - (\alpha'+\beta')x + \alpha'\beta'] = 0\). Let \(k=a\) to match the coefficient of \(x^2\) in the options. New equation: \(a\left[x^2 - \left(-\frac{\sqrt{5}b}{a}\right)x + \left(\frac{5c}{a}\right)\right] = 0\) \(a\left[x^2 + \frac{\sqrt{5}b}{a}x + \frac{5c}{a}\right] = 0\) Multiply by \(a\): \(ax^2 + \sqrt{5}bx + 5c = 0\). This matches option (a). \[ \boxed{ax^2 + \sqrt{5}bx + 5c = 0} \]