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The problem asks to find a quadratic equation whose roots are \(\alpha^2\) and \(\beta^2\), given that \(\alpha\) and \(\beta\) are the roots of the equation:

\[
2x^2 - 3x - 5 = 0.
\]

### Step 1: Recall Vieta's formulas
For the quadratic equation \(ax^2 + bx + c = 0\), the sum and product of roots are:
\[
\text{Sum of roots} = -\frac{b}{a}, \quad \text{Product of roots} = \frac{c}{a}.
\]
Here, \(a = 2\), \(b = -3\), and \(c = -5\). So:
\[
\alpha + \beta = -\frac{-3}{2} = \frac{3}{2}, \quad \alpha\beta = \frac{-5}{2}.
\]

### Step 2: Find the sum and product of \(\alpha^2\) and \(\beta^2\)
The new roots are \(\alpha^2\) and \(\beta^2\). Using the formulas:
- \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\),
- \(\alpha^2 \beta^2 = (\alpha \beta)^2\).

Substitute the known values:
\[
\alpha^2 + \beta^2 = \left(\frac{3}{2}\right)^2 - 2\left(\frac{-5}{2}\right) = \frac{9}{4} + 5 = \frac{29}{4}.
\]
\[
\alpha^2 \beta^2 = \left(\frac{-5}{2}\right)^2 = \frac{25}{4}.
\]

### Step 3: Form the quadratic equation
The quadratic equation with roots \(\alpha^2\) and \(\beta^2\) is:
\[
x^2 - (\alpha^2 + \beta^2)x + \alpha^2\beta^2 = 0.
\]
Substitute \(\alpha^2 + \beta^2 = \frac{29}{4}\) and \(\alpha^2\beta^2 = \frac{25}{4}\):
\[
x^2 - \frac{29}{4}x + \frac{25}{4} = 0.
\]

Multiply through by 4 to eliminate the fractions:
\[
4x^2 - 29x + 25 = 0.
\]

### Final Answer:
The quadratic equation is:
\[
4x^2 - 29x + 25 = 0.
\]

Would you like further clarification or more examples? Here are 5 related questions you might find interesting:

1. How do Vieta's formulas apply to higher-degree polynomials?
2. Can we generalize this process for roots that are cubes (\(\alpha^3, \beta^3\))?
3. How would the equation change if the roots were \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\)?
4. What happens if \(\alpha\) and \(\beta\) are complex roots?
5. How can we verify the solution graphically or numerically?

**Tip:** Always simplify expressions involving the sum and product of roots systematically to avoid algebraic errors!

If α, β are roots of equation 2x^2 - 3x - 5 = 0, then find a quadratic equation whose roots are α^2, β^2.

Learn how to derive a quadratic equation with roots α^2 and β^2, given that α and β are roots of the equation 2x^2 - 3x - 5 = 0. Explore key concepts like Vieta's formulas, sum and product of roots, and algebraic manipulation. Suitable for Grades 10-12.

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