Question

The quadratic equation whose roots are three times the roots of the equation \( 2x^2 + 3x + 5 = 0 \), is

• A. \( 2x^2 + 9x + 45 = 0 \)
• B. \( 2x^2 + 9x - 45 = 0 \)
• C. \( 5x^2 + 9x + 45 = 0 \)
• D. \( 2x^2 - 9x + 45 = 0 \)
• E. \( 2x^2 + 9x + 49 = 0 \)

Hint:

To scale roots by \(k\), substitute \(x \to x/k\) and simplify.

Answer 1

The Correct Option is A

Understanding the Concept: If roots are multiplied by \(k\), replace \(x\) by \(x/k\).

Step 1: Given equation
\[ 2x^2 + 3x + 5 = 0 \]

Step 2: Replace \(x \to x/3\)
\[ 2\left(\frac{x}{3}\right)^2 + 3\left(\frac{x}{3}\right) + 5 = 0 \]

Step 3: Simplify
\[ 2\frac{x^2}{9} + x + 5 = 0 \] \[ \frac{2x^2}{9} + x + 5 = 0 \]

Step 4: Multiply by 9
\[ 2x^2 + 9x + 45 = 0 \]

Step 5: Final Answer
\[ \boxed{2x^2 + 9x + 45 = 0} \]