1. The given expression $\frac{-b\sqrt{b^2-4ac}}{2a}$ is not quite the correct formula for finding the roots of a quadratic function. The correct quadratic formula for finding the roots of the equation $ax^2 + bx + c = 0$ is:

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

The term inside the square root, $b^2 - 4ac$, is called the discriminant. It determines the nature of the roots:

• If $b^2 - 4ac > 0$, there are two distinct real roots.
• If $b^2 - 4ac = 0$, there is one real root (a repeated root).
• If $b^2 - 4ac < 0$, there are two complex roots.

2. To find the intercepts and intersection points of a parabola given by the quadratic equation $ax^2 + bx + c$, follow these steps:

Finding the Intercepts

Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. This occurs when $x = 0$. Substitute $x = 0$ into the quadratic equation to find the y-coordinate:

$y = a(0)^2 + b(0) + c = c$

So, the y-intercept is $(0, c)$.

X-intercepts (Roots): The x-intercepts are the points where the parabola intersects the x-axis. These occur where $y = 0$. To find the x-intercepts, solve the quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula:

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

These solutions give the x-coordinates of the intercepts.

Finding the Vertex

The vertex of the parabola is given by the formula $x = -\frac{b}{2a}$. To find the y-coordinate of the vertex, substitute this x-value back into the original equation:

$y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c$

Simplify to get the y-coordinate of the vertex.

Finding the Intersection with Another Line

To find the intersection points of the parabola with another line given by $y = mx + b$, substitute $y = mx + b$ into the quadratic equation:

$ax^2 + bx + c = mx + b$

Rearrange to form a new quadratic equation:

$ax^2 + (b - m)x + (c - b) = 0$

Solve this new quadratic equation using the quadratic formula to find the x-coordinates of the intersection points. Substitute these x-values back into $y = mx + b$ to find the corresponding y-coordinates.

Would you like more details or have any questions? Here are five questions you might want to ask:

1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in the quadratic formula?
3. Can you explain the steps to complete the square for solving a quadratic equation?
4. How can you determine the axis of symmetry for a parabola?
5. What are some practical applications of quadratic equations?

Tip: Always check the discriminant first when solving quadratic equations to understand the nature of the roots before applying the quadratic formula.