To find the quadratic equation with roots $\frac{1}{p^2}$ and $\frac{1}{q^2}$, let's proceed with the following steps:

Step 1: Sum and Product of Original Roots

For the original quadratic equation $ax^2 + bx + c = 0$, with roots $p$ and $q$, we know:

1. Sum of the roots $p + q = -\frac{b}{a}$
2. Product of the roots $pq = \frac{c}{a}$

Step 2: Determine the Sum and Product of New Roots

The new roots are $\frac{1}{p^2}$ and $\frac{1}{q^2}$.

1. Sum of new roots: $\frac{1}{p^2} + \frac{1}{q^2} = \frac{p^2 + q^2}{p^2 q^2}$ Using the identity $p^2 + q^2 = (p + q)^2 - 2pq$, we get: $p^2 + q^2 = \left(-\frac{b}{a}\right)^2 - 2 \cdot \frac{c}{a} = \frac{b^2 - 2ac}{a^2}$ Therefore, $\frac{1}{p^2} + \frac{1}{q^2} = \frac{\frac{b^2 - 2ac}{a^2}}{\left(\frac{c}{a}\right)^2} = \frac{b^2 - 2ac}{c^2}$

2. Product of new roots: $\frac{1}{p^2} \cdot \frac{1}{q^2} = \frac{1}{(pq)^2} = \frac{1}{\left(\frac{c}{a}\right)^2} = \frac{a^2}{c^2}$

Step 3: Form the New Quadratic Equation

A quadratic equation with roots $\alpha$ and $\beta$ can be written as: $x^2 - (\text{sum of roots}) \cdot x + (\text{product of roots}) = 0$

Substituting the values for our new roots: $x^2 - \left(\frac{b^2 - 2ac}{c^2}\right)x + \frac{a^2}{c^2} = 0$

Final Answer

The required quadratic equation whose roots are $\frac{1}{p^2}$ and $\frac{1}{q^2}$ is: $c^2 x^2 - (b^2 - 2ac)x + a^2 = 0$

Would you like further clarification on any step?

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1. How do we find the roots of a transformed quadratic equation?
2. Why is the product of roots important in forming a new quadratic equation?
3. How can we verify roots without solving the entire equation?
4. What are other ways to manipulate polynomial roots for transformation?
5. How does changing roots impact the coefficients of the new polynomial?

Tip: When transforming a quadratic equation, always check the sum and product of the new roots for consistency.