https://r2.erweima.ai/mathbot/image/1726517033014ihlgk7b6.jpg

Let's solve the quadratic equation provided in the image:

The equation is: 

\[
(4 - \alpha) x^2 + 2 (\alpha x + 1) = 0
\]

First, expand the equation:

\[
(4 - \alpha) x^2 + 2\alpha x + 2 = 0
\]

For this quadratic equation to have a unique solution, the discriminant (denoted as \(\Delta\)) must be equal to zero. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

where \(a = 4 - \alpha\), \(b = 2\alpha\), and \(c = 2\).

The discriminant \(\Delta\) is given by:

\[
\Delta = b^2 - 4ac
\]

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

\[
\Delta = (2\alpha)^2 - 4(4 - \alpha)(2)
\]

Simplify the equation:

\[
\Delta = 4\alpha^2 - 4(4 - \alpha)(2)
\]
\[
\Delta = 4\alpha^2 - 8(4 - \alpha)
\]
\[
\Delta = 4\alpha^2 - 32 + 8\alpha
\]
\[
\Delta = 4\alpha^2 + 8\alpha - 32
\]

Since the equation has a unique solution, \(\Delta = 0\):

\[
4\alpha^2 + 8\alpha - 32 = 0
\]

Now, solve this quadratic equation:

\[
\alpha^2 + 2\alpha - 8 = 0
\]

We can solve this quadratic equation using the quadratic formula:

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

where \(a = 1\), \(b = 2\), and \(c = -8\). Plugging these values into the formula:

\[
\alpha = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)}
\]
\[
\alpha = \frac{-2 \pm \sqrt{4 + 32}}{2}
\]
\[
\alpha = \frac{-2 \pm \sqrt{36}}{2}
\]
\[
\alpha = \frac{-2 \pm 6}{2}
\]

Thus, the solutions for \(\alpha\) are:

\[
\alpha = \frac{-2 + 6}{2} = 2 \quad \text{or} \quad \alpha = \frac{-2 - 6}{2} = -4
\]

Therefore, the possible values of \(\alpha\) are \( \alpha = 2 \) or \( \alpha = -4 \).

Would you like further details or explanations? Here are five related questions to expand on this:

1. How does the discriminant determine the number of solutions for a quadratic equation?
2. What happens if the discriminant is greater than zero in a quadratic equation?
3. Could you verify the solution by substituting \(\alpha = 2\) and \(\alpha = -4\) back into the original equation?
4. How would the solution process change if there were two distinct solutions instead of one?
5. What are some other methods for solving quadratic equations besides the quadratic formula?

**Tip**: When solving quadratic equations, always check whether factoring, completing the square, or using the quadratic formula is the most efficient method!

Calcula el valor de α si la ecuación de segundo grado: (4 - α)x^2 + 2(αx + 1) = 0 tiene solución única.

Learn how to solve the quadratic equation (4 - α)x^2 + 2(αx + 1) = 0 for a unique solution by using the discriminant. This problem explains key algebraic concepts like quadratic equations, the discriminant formula, and solving for α. Suitable for students in Grades 10-12.

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