91Ó°ÊÓ

To find the x-intercepts of a quadratic function, we can use the quadratic formula. The quadratic formula is a powerful tool that helps solve any quadratic equation of the form:
\[ ax^2 + bx + c = 0 \]
Using the quadratic formula, \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] you can find the roots (solutions) of the quadratic equation. These roots or solutions are the x-values where the function intersects the x-axis.
In our case, the given quadratic function is: \[ f(x) = x^2 - 3x + 10 \].
Here, the coefficients are

• \( a = 1 \)
• \( b = -3 \)
• \( c = 10 \)

Plugging these values into the quadratic formula helps us determine the x-intercepts. However, relying solely on the formula without considering the discriminant might lead to confusion. We should use discriminant analysis to fully understand the nature of the solutions.

The discriminant is part of the quadratic formula under the square root:
\[ \text{Discriminant} = b^2 - 4ac \]
The discriminant tells us about the nature of the roots of the quadratic equation:

• If \( \text{Discriminant} > 0 \), the equation has two distinct real roots.
• If \( \text{Discriminant} = 0 \), the equation has exactly one real root (a repeated root).
• If \( \text{Discriminant} < 0 \), the equation has no real roots (the solutions are complex or imaginary).

For our function \[ f(x) = x^2 - 3x + 10 \],
plugging in the coefficients into the discriminant formula, we find: \[ \text{Discriminant} = (-3)^2 - 4(1)(10) = 9 - 40 = -31 \].
Since the discriminant is negative \( (\text{Discriminant} < 0) \), this tells us that the quadratic equation has no real roots or x-intercepts. The solutions are complex numbers. Understanding this helps us predict the behavior of the parabola on a graph.

The nature of the roots from the discriminant analysis can help us predict the behavior of a parabola. In our problem, the negative discriminant means that the parabola does not intersect the x-axis at any point.
A parabola is defined by its vertex and the direction in which it opens. The quadratic function \( f(x) = x^2 - 3x + 10 \) opens upwards because the coefficient of \( x^2 \) (a = 1) is positive. This also means that the vertex of the parabola is the lowest point on the graph.
Since there are no x-intercepts, the vertex lies entirely above the x-axis, ensuring the function does not touch or cross the x-axis at any value.
Understanding the behavior of parabolas helps in visualizing and graphing quadratic functions. Keep in mind:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.
• The vertex can be found using the formula \( x = -\frac{b}{2a} \).