For a quadratic equation in the standard form \(ax^2 + bx + c = 0\), the x-intercepts (also known as the roots or solutions) are found using the quadratic formula:

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

The x-intercepts are the points where the graph of the quadratic equation intersects the x-axis, which means these are the values of \(x\) for which \(y = 0\).

Here's a step-by-step process to find the x-intercepts using the quadratic formula:

1. **Identify the coefficients**: From the equation \(ax^2 + bx + c = 0\), identify \(a\), \(b\), and \(c\).
2. **Compute the discriminant**: The discriminant is \(\Delta = b^2 - 4ac\).
   - If \(\Delta > 0\), there are two distinct real roots (x-intercepts).
   - If \(\Delta = 0\), there is exactly one real root (x-intercept).
   - If \(\Delta < 0\), there are no real roots (the quadratic does not intersect the x-axis).
3. **Apply the quadratic formula**: Plug \(a\), \(b\), and \(\Delta\) into the quadratic formula:

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

4. **Solve for \(x\)**: Calculate the two possible values for \(x\) (if \(\Delta \geq 0\)).

### Example

Consider the quadratic equation \(2x^2 - 4x - 6 = 0\):

1. **Identify the coefficients**: \(a = 2\), \(b = -4\), \(c = -6\).
2. **Compute the discriminant**: \(\Delta = (-4)^2 - 4(2)(-6) = 16 + 48 = 64\).
3. **Apply the quadratic formula**:

   \[ x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4} \]

4. **Solve for \(x\)**:

   \[ x_1 = \frac{4 + 8}{4} = 3 \]
   \[ x_2 = \frac{4 - 8}{4} = -1 \]

So, the x-intercepts are \(x = 3\) and \(x = -1\).

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the vertex of a quadratic function?
2. What is the axis of symmetry of a quadratic function?
3. How does the discriminant determine the nature of the roots of a quadratic equation?
4. Can a quadratic equation have no x-intercepts? Why or why not?
5. How do you complete the square to solve a quadratic equation?
6. What is the difference between the standard form and the vertex form of a quadratic equation?
7. How can you determine if a quadratic function opens upwards or downwards?
8. What is the significance of the coefficients \(a\), \(b\), and \(c\) in a quadratic equation?

**Tip:** Always check the discriminant before solving a quadratic equation to understand the nature of the roots you will get. This helps in predicting the number of real solutions the equation has.

Explore how to solve quadratic equations using the quadratic formula. Learn step-by-step methods, examples, and the significance of discriminants. Perfect for high school students studying algebra and quadratic functions.

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