To solve this problem, let us break it down step by step.

Given: $f(x) = x^2 + 2x + 9$ This is a quadratic equation, and we know that $\alpha$ and $\beta$ are the roots of this quadratic equation. We need to find the quadratic equation whose roots are $\alpha^2 - 1$ and $\beta^2 - 1$.

Step 1: Use Vieta's Formulas for the original quadratic equation

For the quadratic equation $f(x) = x^2 + 2x + 9$, we can use Vieta's formulas to relate the sum and product of the roots:

• The sum of the roots $\alpha + \beta$ is given by the opposite of the coefficient of $x$, i.e., $\alpha + \beta = -\frac{2}{1} = -2$
• The product of the roots $\alpha \beta$ is the constant term divided by the leading coefficient, i.e., $\alpha \beta = \frac{9}{1} = 9$

Step 2: Define the new roots and express the sum and product

Now, we need to find the quadratic equation whose roots are $\alpha^2 - 1$ and $\beta^2 - 1$. Let's call these new roots $p = \alpha^2 - 1$ and $q = \beta^2 - 1$. The new quadratic equation will have the form:

$x^2 - (p + q)x + pq = 0$

Step 2a: Find the sum of the new roots, $p + q$

We know that: $p = \alpha^2 - 1 \quad \text{and} \quad q = \beta^2 - 1$ Thus: $p + q = (\alpha^2 - 1) + (\beta^2 - 1) = \alpha^2 + \beta^2 - 2$

To simplify $\alpha^2 + \beta^2$, we use the identity: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$ Substituting the known values: $\alpha^2 + \beta^2 = (-2)^2 - 2(9) = 4 - 18 = -14$ Therefore: $p + q = -14 - 2 = -16$

Step 2b: Find the product of the new roots, $pq$

The product of the new roots is: $pq = (\alpha^2 - 1)(\beta^2 - 1)$ We can expand this expression: $pq = \alpha^2 \beta^2 - \alpha^2 - \beta^2 + 1$ Since $\alpha^2 \beta^2 = (\alpha \beta)^2 = 9^2 = 81$, and we already know $\alpha^2 + \beta^2 = -14$, we can substitute: $pq = 81 - (-14) + 1 = 81 + 14 + 1 = 96$

Step 3: Form the new quadratic equation

Now that we know the sum and product of the new roots, we can form the new quadratic equation: $x^2 - (p + q)x + pq = x^2 - (-16)x + 96 = x^2 + 16x + 96$

Final Answer:

The quadratic equation whose roots are $\alpha^2 - 1$ and $\beta^2 - 1$ is: $x^2 + 16x + 96 = 0$

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Would you like any more details or clarifications on the solution?

Here are some follow-up questions for you:

1. How do you use Vieta's formulas to find the sum and product of roots?
2. What would happen if the given quadratic equation had complex roots?
3. How could you modify the method if the roots were transformed differently, such as $\alpha^2 + 1$ and $\beta^2 + 1$?
4. Can this process be applied to higher-degree polynomials? If so, how?
5. How can you calculate the discriminant of the quadratic equation you just found?

Tip: When solving problems involving roots of quadratics, always remember to use Vieta's formulas as a quick way to find relationships between the roots.