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The given equation is:

\[
x^2 - 2x - 35 = 0
\]

We are asked to find \( \alpha^2 + \beta \), where \( \alpha \) and \( \beta \) are the solutions of the equation, and \( \alpha > \beta \).

### Step 1: Solving for \( \alpha \) and \( \beta \)

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -2 \), and \( c = -35 \).

We can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-35)}}{2(1)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 140}}{2}
\]
\[
x = \frac{2 \pm \sqrt{144}}{2}
\]
\[
x = \frac{2 \pm 12}{2}
\]

So the two solutions are:

\[
x_1 = \frac{2 + 12}{2} = 7 \quad \text{and} \quad x_2 = \frac{2 - 12}{2} = -5
\]

Thus, \( \alpha = 7 \) and \( \beta = -5 \).

### Step 2: Calculating \( \alpha^2 + \beta \)

We are asked to find \( \alpha^2 + \beta \). Substituting the values of \( \alpha \) and \( \beta \):

\[
\alpha^2 + \beta = 7^2 + (-5) = 49 - 5 = 44
\]

Therefore, the value of \( \alpha^2 + \beta \) is \( \boxed{44} \).

Would you like a more detailed explanation of the quadratic formula or any other part of the solution? Here are five related questions you might be interested in:

1. What is the derivation of the quadratic formula?
2. How can you verify solutions of a quadratic equation by factoring?
3. What is the significance of the discriminant \( \Delta \) in quadratic equations?
4. How do you solve quadratic equations using the method of completing the square?
5. What is the relationship between the roots and coefficients of a quadratic equation?

**Tip**: For any quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots is \( -\frac{b}{a} \) and the product of the roots is \( \frac{c}{a} \).

Sea α y β las soluciones de la siguiente ecuación x^2 - 2x - 35 = 0, si α > β. Calcule α^2 + β

Learn how to solve the quadratic equation x² - 2x - 35 = 0 where α > β. This step-by-step solution explains how to apply the quadratic formula and calculate α² + β. Suitable for students in Grades 9-10.

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