y = x^2 - 3x - 2 solve for x and determine if it's a function

To solve the quadratic equation \( y = x^2 - 3x - 2 \) for \( x \), we'll set \( y = 0 \) and solve the resulting quadratic equation:

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

This equation can be solved using the quadratic formula:

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

For the equation \( x^2 - 3x - 2 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -3 \)
- \( c = -2 \)

Substituting these values into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-2)}}{2(1)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 8}}{2}
\]
\[
x = \frac{3 \pm \sqrt{17}}{2}
\]

So, the solutions for \( x \) are:

\[
x_1 = \frac{3 + \sqrt{17}}{2}, \quad x_2 = \frac{3 - \sqrt{17}}{2}
\]

### Is it a function?
Yes, the equation \( y = x^2 - 3x - 2 \) represents a function. Specifically, it is a quadratic function, and for each value of \( x \), there is exactly one corresponding value of \( y \). This ensures that it satisfies the definition of a function, which requires that each input (value of \( x \)) has a single output (value of \( y \)).

Would you like more details or have any questions?

Here are some related questions:
1. How do you determine the vertex of the quadratic function \( y = x^2 - 3x - 2 \)?
2. How do you graph the function \( y = x^2 - 3x - 2 \)?
3. What are the axis of symmetry and the direction of the parabola for this function?
4. How do you find the discriminant of a quadratic equation, and what does it indicate about the roots?
5. Can this function be factored directly, and if so, how?

**Tip:** When solving quadratic equations, the discriminant (\(b^2 - 4ac\)) is crucial in determining the nature of the roots—whether they are real or complex.

Learn how to solve the quadratic equation y = x^2 - 3x - 2 step-by-step and determine if it represents a function. This problem covers quadratic equations, the quadratic formula, and the definition of functions in mathematics. Suitable for high school students and anyone preparing for algebraic functions.

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