Question

The graph of a function of the form ​f(x)equalsax squared plus bx plus c for different values of​ a, b, and c is given. For the​ function, find the following.
​(a) Determine if the discriminant is​ positive, negative, or zero.
​(b) Determine if there are​ 0, 1, or 2 real solutions to ​f(x)equals0.
​(c) Solve the equation ​f(x)equals0.
-10
-5
5
10
4
-12
-24
8
x
y



A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 24 to 8 in increments of 2. A parabola that opens upward has vertex at approximately (1, negative 9) and passes through the points (negative 2, 0) and (4, 0).
Question content area bottom
Part 1
​(a) Choose the correct answer below.
A.
The discriminant of the equation is negative.
B.
The discriminant of the equation is zero.
C.
The discriminant of the equation is positive.

Answer 1

To analyze the quadratic function \( f(x) = ax^2 + bx + c \) given the information about its graph, let’s address each of the parts of the question.

### Part (a): Determine the Discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[
D = b^2 - 4ac
\]

The discriminant tells us about the nature of the roots:
- \( D > 0 \) implies there are 2 distinct real solutions.
- \( D = 0 \) implies there is exactly 1 real solution (a repeated root).
- \( D < 0 \) implies there are no real solutions (the roots are complex).

From the provided description, the parabola opens upwards and has zeros (real roots) at \( x = -2 \) and \( x = 4 \). Since it has two real intercepts on the x-axis, we can conclude that the discriminant is positive.

So the answer to Part (a) is:
**C. The discriminant of the equation is positive.**

### Part (b): Determine the Number of Real Solutions
Given that the discriminant is positive (from our analysis in Part (a)), there are 2 distinct real solutions to the equation \( f(x) = 0 \).

So the answer to Part (b) is:
**There are 2 real solutions to \( f(x) = 0 \).**

### Part (c): Solve the Equation \( f(x) = 0 \)
The solutions can be obtained by solving the equation \( f(x) = 0 \), which, based on the roots identified earlier, can be factored as:
\[
f(x) = a(x + 2)(x - 4) = 0
\]

The roots are:
1. \( x + 2 = 0 \rightarrow x = -2 \)
2. \( x - 4 = 0 \rightarrow x = 4 \)

So the solutions to \( f(x) = 0 \) are:
\[
x = -2 \quad \text{and} \quad x = 4
\]

In summary:
- For Part (a), the discriminant is positive.
- For Part (b), there are 2 real solutions.
- For Part (c), the solutions to \( f(x) = 0 \) are \( x = -2 \) and \( x = 4 \).