Find the roots and use them to graph the related function.
{ x| - 0.5 ≤ x ≤ 2.5 }
Practice makes perfect
We will start by rearranging the given quadratic inequality.
0 ≤ - 4x^2+8x+5 ⇔ - 4x^2+8x+5 ≥ 0
We will now sketch the related quadratic function. To do so, we first need to identify the values of a, b, and c.
y= - 4x^2 + 8x + 5
We see that a= - 4, b= 8, and c= 5. Let's substitute these values into the Quadratic Formula to find the roots of - 2x^2+4x+4=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=- 8 ± 12/- 8
x=- 8 + 12/- 8
x=- 8 - 12/- 8
x=4/- 8
x=- 20/- 8
x= - 0.5
x= 2.5
The solution of the given quadratic inequality, - 4x^2+8x+5 ≥ 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= - 4 is less than zero.
We see that the graph lies on and above the x-axis between x ≥ - 0.5 and x ≤ 2.5.
{ x| - 0.5 ≤ x ≤ 2.5 }
⇕
[- 0.5,2.5]
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around - 0.5 and 2.5. 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= - 4 and b= 8 into - b2a.
