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.