Question

question 2
1 pts
use your calculator to find the solutions and the maximum or minimum point of the quadratic equation.
x² + 9x + 20
○ x = 4, x = 5; maximum (-4.5, -0.25)
○ x = -4, x = -5; minimum (-4.0, -1.25)
○ x = 4, x = 5; maximum (-4.0, -1.25)
○ x = -4, x = -5; minimum (-4.5, -0.25)

Explanation:

Step1: Solve the quadratic equation

To find the solutions of \(x^2 + 9x + 20 = 0\), we factor the quadratic. We need two numbers that multiply to \(20\) and add to \(9\). Those numbers are \(4\) and \(5\), so we can factor it as \((x + 4)(x + 5)=0\). Setting each factor equal to zero gives \(x + 4 = 0\) or \(x + 5 = 0\), so \(x=-4\) or \(x = -5\).

Step2: Determine the vertex (maximum or minimum)

For a quadratic function \(y = ax^2+bx + c\) (here \(a = 1\), \(b = 9\), \(c = 20\)), the x - coordinate of the vertex is given by \(x=-\frac{b}{2a}\). Substituting \(a = 1\) and \(b = 9\), we get \(x=-\frac{9}{2(1)}=-4.5\).
To find the y - coordinate, substitute \(x=-4.5\) into the function: \(y=(-4.5)^2+9(-4.5)+20\).
First, \((-4.5)^2 = 20.25\), \(9(-4.5)=-40.5\). Then \(y = 20.25-40.5 + 20=20.25 + 20-40.5=40.25-40.5=-0.25\).
Since \(a = 1>0\), the parabola opens upwards, so the vertex is a minimum point.

Answer:

x = -4, x = -5; Minimum (-4.5, -0.25)