find x and y intecepts in y = ax squared + bx + c

To find the intercepts of the quadratic equation \( y = ax^2 + bx + c \), we will look for both the **x-intercepts** and the **y-intercept**.

### 1. **Y-Intercept:**

The y-intercept occurs where the value of \( x = 0 \). To find it, substitute \( x = 0 \) into the equation:

\[
y = a(0)^2 + b(0) + c = c
\]

Thus, the **y-intercept** is at the point \( (0, c) \).

### 2. **X-Intercepts:**

The x-intercepts occur where the value of \( y = 0 \). Set \( y = 0 \) and solve for \( x \) in the equation:

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

This is a quadratic equation, and you can solve for \( x \) using the **quadratic formula**:

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

The solutions to this equation (if real) will give you the **x-intercepts**. These intercepts are the points \( \left( \frac{-b + \sqrt{b^2 - 4ac}}{2a}, 0 \right) \) and \( \left( \frac{-b - \sqrt{b^2 - 4ac}}{2a}, 0 \right) \).

If the discriminant \( (b^2 - 4ac) \) is:
- Positive: You get two distinct real x-intercepts.
- Zero: You get one real x-intercept (the vertex of the parabola).
- Negative: There are no real x-intercepts, meaning the parabola does not cross the x-axis.

### Summary:
- **Y-intercept**: \( (0, c) \)
- **X-intercepts**: Use the quadratic formula to solve \( ax^2 + bx + c = 0 \) if real solutions exist.

Would you like more details on solving for the x-intercepts?

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Here are 5 related questions to explore further:
1. How do you derive the quadratic formula?
2. What happens if \( a = 0 \) in the equation \( y = ax^2 + bx + c \)?
3. Can you always find real x-intercepts for any quadratic equation?
4. What is the significance of the discriminant \( b^2 - 4ac \) in determining the shape of a parabola?
5. How do x- and y-intercepts relate to the graph of a quadratic function?

**Tip**: The sign of \( a \) in a quadratic function \( y = ax^2 + bx + c \) determines if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).

find x and y intercepts in y = ax squared + bx + c

Discover how to find the x- and y-intercepts for a quadratic equation of the form y = ax^2 + bx + c. This guide covers essential algebraic concepts, including the quadratic formula and discriminant, perfect for students in Grades 9-10.

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