b) We know that the quadratic formula can be used to determine the roots of a quadratic function. However, the quadratic formula also reveals how many roots exist without the need to find them. Explain why (not how) we are able to determine the number of roots a quadratic has without actually determining what they are.
Please answer each question by indicating which question you are answering.
Ex: Part A
A) Answer here
Thank you!
oobleck
answered
so the axis of symmetry is at x = -b/2a, midway between the roots, if any.
If b^2-4ac is negative, there are no real roots. Thus the name "discriminant," because it allows you to discriminate the number and type of roots.
Step-by-Step Bot
answered
The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two symmetric halves. It is represented by the equation x = -b/2a, where a, b, and c are coefficients of the quadratic equation in the form ax^2 + bx + c = 0. The quadratic formula is used to find the x-intercepts or roots of the quadratic equation. The axis of symmetry is related to the quadratic formula because the x-coordinate of the vertex of the parabola, which lies on the axis of symmetry, can be found using the quadratic formula.
Part B:
The quadratic formula, when used to find the roots of a quadratic equation, uses the discriminant (b^2 - 4ac) to determine the nature of the solutions. The discriminant indicates the number of roots and their nature without actually determining what they are.
If the discriminant is positive (b^2 - 4ac > 0), the quadratic equation has two distinct real roots.
If the discriminant is zero (b^2 - 4ac = 0), the quadratic equation has one real root (also called a double root). In this case, the parabola touches the x-axis at one point.
If the discriminant is negative (b^2 - 4ac < 0), the quadratic equation has no real roots. This means that the parabola does not intersect the x-axis and remains above or below it in relation to the y-axis.
Therefore, by examining the discriminant of a quadratic equation, we can determine the number of distinct real roots, the presence of one real root (double root), or the absence of real roots, without having to actually find the roots themselves.
Explain Bot
answered
The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two symmetric halves. It passes through the vertex of the quadratic function. On the other hand, the quadratic formula is a mathematical equation used to find the roots of a quadratic function. It states that the roots of a quadratic function can be calculated using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Now, let's see how these two concepts are related. The axis of symmetry of a quadratic function can be represented by the equation x = h, where h is the x-coordinate of the vertex of the parabola. In the general quadratic function form, f(x) = ax^2 + bx + c, with a, b, and c being constants, the x-coordinate of the vertex is given by h = -b/(2a).
By comparing the equation x = h with the quadratic formula, we can observe that the x-coordinate of the vertex (-b/(2a)) is in fact the value of x that yields the maximum or minimum point of the parabola. Therefore, the axis of symmetry and the quadratic formula provide information about the x-coordinate of the vertex, allowing us to find the roots of a quadratic function.
Part B:
The quadratic formula, which is x = (-b ± √(b^2 - 4ac))/(2a), provides a way to determine the roots of a quadratic function. The discriminant, which is the value inside the square root, b^2 - 4ac, plays a crucial role in determining the number of roots.
If the discriminant is positive (b^2 - 4ac > 0), then the quadratic function has two distinct real roots. Each root corresponds to one of the ± signs in the quadratic formula.
If the discriminant is zero (b^2 - 4ac = 0), then the quadratic function has exactly one real root. In this case, the ± sign in the quadratic formula becomes irrelevant, as both the added and subtracted versions yield the same value.
If the discriminant is negative (b^2 - 4ac < 0), then the quadratic function has no real roots. This is because the square root of a negative number is not a real number.
By examining the value of the discriminant, we can determine whether the quadratic function has zero, one, or two real roots without actually finding the roots themselves. This information helps us understand the behavior and properties of the quadratic equation.
