Question

question 7 (mandatory) (1 point) saved
all of the following are ways to determine the roots of a quadratic function except:

a) graphing

b) completing the square

c) quadratic formula

d) finding the discriminant

question 8 (mandatory) (1 point) saved
the population of a village can be modelled by the function $p(x) = -22.5x^2 + 428x + 1100$, where $x$ is the number of years since 1990. according to the model, when will the population be the highest?

Explanation:

Question 7

To determine the roots of a quadratic function:

• Graphing: The roots are the x - intercepts of the parabola, so this is a valid method.
• Completing the square: This algebraic method can be used to solve for the values of \(x\) (the roots) by rewriting the quadratic in a perfect - square form.
• Quadratic formula: The quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) is a direct way to find the roots of a quadratic equation \(ax^{2}+bx + c = 0\).
• Finding the discriminant: The discriminant (\(D = b^{2}-4ac\)) tells us about the nature of the roots (whether they are real, equal, or complex) but does not give the actual roots themselves. So finding the discriminant is not a way to determine the roots.

Step 1: Recall the formula for the vertex of a parabola

For a quadratic function in the form \(f(x)=ax^{2}+bx + c\) (where \(a
eq0\)), the x - coordinate of the vertex (which gives the time when the maximum or minimum value of the function occurs) is given by \(x =-\frac{b}{2a}\). Since the coefficient of \(x^{2}\) (\(a=- 22.5\)) is negative, the parabola opens downwards, and the vertex represents the maximum point of the function.

Step 2: Identify the values of \(a\) and \(b\)

In the function \(P(x)=-22.5x^{2}+428x + 1100\), we have \(a=-22.5\) and \(b = 428\).

Step 3: Calculate the x - coordinate of the vertex

Substitute \(a=-22.5\) and \(b = 428\) into the formula \(x=-\frac{b}{2a}\).

$$
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$$

Step 4: Find the year

Since \(x\) is the number of years since 1990, we add \(x\) to 1990. So the year is \(1990 + 9.51\approx1999.51\), which is approximately the year 2000 (or 1999 - 2000, more precisely around 1999.5).

Answer:

d) finding the discriminant

Question 8