We will start by sketching the related quadratic function. To do so, we first need to identify the values of a, b, and c.
y= - 2x^2 + 3x + 3
We see that a= - 2, b= 3, and c= 3. Let's substitute these values into the Quadratic Formula to find the roots of - 2x^2+3x+3=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=- 3 ± sqrt(33)/- 4
x=- 3 + sqrt(33)/- 4
x=- 3 - sqrt(33)/- 4
x=3/4-sqrt(33)/4
x=3/4+sqrt(33)/4
x≈ - 0.69
x≈ 2.19
The solution of the given quadratic inequality, -2x^2+3x+3≤0, consists of x-values for which the graph of the related quadratic function lies on and below the x-axis. The graph opens downward, since a= - 2 is less than zero.
We see that the graph lies on and below the x-axis at about x≤ - 0.69 and x ≥ 2.19.
{ x| x ≤ - 0.69 or x ≥ 2.19 }
⇕
(- ∞, - 0.69] ⋃ [2.19, ∞ )
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around - 0.69 and 2.19. To draw the graph, we will find the vertex, (- b2a,f(- b2a)). Let's start by calculating its x-coordinate. To do so, we will substitute a= - 2 and b= 3 into - b2a.
