The solutions to nx^2 + px + q=0 are 4 and -2/5. What information does this tell you about the graph of f(x)= nx^2 + px +q?

If you could answer this for me, that would be great. Thanks in advance :)

all it tells me is that it is a parabola that crosses the x-axis at 4 and -2/5.

The solutions to the quadratic equation nx^2 + px + q = 0, given as 4 and -2/5, provide information about the roots or x-intercepts of the quadratic function f(x) = nx^2 + px + q.

1. The fact that the solutions are 4 and -2/5 indicates that the quadratic equation has two distinct roots. This means that the graph of the quadratic function f(x) will intersect the x-axis at two different points.

2. The first solution 4 suggests that the graph of f(x) intersects the x-axis at x = 4. This means that when x is equal to 4, the value of the function f(x) is 0.

3. The second solution -2/5 tells us that the graph of f(x) intersects the x-axis at x = -2/5. This means that when x is equal to -2/5, the value of the function f(x) is also 0.

4. Based on the given solutions and the nature of quadratic functions, we can expect the graph of f(x) to be a parabola that opens upward or downward, depending on the values of n. The vertex of the parabola, which represents the minimum or maximum point of the function, will lie on the line of symmetry between the x-intercepts.

Overall, the solutions 4 and -2/5 inform us about the x-intercepts, or roots, of the quadratic equation, and give insight into the shape and position of the graph of f(x) = nx^2 + px + q.

To understand what this information tells us about the graph of the quadratic function f(x) = nx^2 + px + q, we can analyze the roots of the equation nx^2 + px + q = 0.

The given equation has two solutions: 4 and -2/5. These solutions represent the x-values at which the quadratic function crosses the x-axis or, in other words, where the function equals zero.

1. Solution x = 4:
When x = 4, substituting this value into the quadratic equation, we get:
n(4)^2 + p(4) + q = 0
16n + 4p + q = 0

2. Solution x = -2/5:
When x = -2/5, substituting this value into the quadratic equation, we get:
n(-2/5)^2 + p(-2/5) + q = 0
4n/25 - 2p/5 + q = 0

By analyzing these two equations, we can draw some conclusions about the graph of the quadratic function:

1. The graph of the function will intersect the x-axis at x = 4 and x = -2/5. These are the x-coordinates of the roots/solutions.
2. Since both solutions are real numbers, this means that the graph will have real x-intercepts.
3. The number of solutions (two in this case) implies that the graph of the quadratic function will intersect the x-axis at two distinct points.
4. The parabola of the quadratic function will open upward if the coefficient n (leading coefficient) is positive, and downward if n is negative.
5. The vertex of the parabola would be positioned between the two x-intercepts (roots).
6. The value q represents the y-coordinate of the vertex, giving the vertical shift of the parabola.
7. The value p, combined with n, affects the horizontal position and width of the parabola. A larger absolute value of p generally gives a wider parabola, while positive or negative p determines the horizontal shift.

Overall, based on the given solutions, we can conclude that the quadratic function will have real and distinct x-intercepts, and the parabola will open either upward or downward based on the value of n. The values of p and q will affect the specific position and shape of the parabola.