Find the roots and use them to graph the related function.
{ x| x < - 5.45 or x > - 0.55 }
Practice makes perfect
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= 1x^2 + 6x + 3
We see that a= 1, b= 6, and c= 3. Let's substitute these values into the Quadratic Formula to find the roots of x^2+6x+3=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=- 6±sqrt(24)/2
x=- 6 + sqrt(24)/2
x=- 6 - sqrt(24)/2
x=- 6/2+sqrt(24)/2
x=- 6/2-sqrt(24)/2
x≈ - 0.55
x≈ - 5.45
The solution of the given quadratic inequality, x^2+6x+3 > 0, consists of x-values for which the graph of the related quadratic function lies above the x-axis. The graph opens upward since a= 1 is greater than zero.
We see that the graph lies above the x-axis at about x < - 5.45 and x > - 0.55.
{ x| x < - 5.45 or x > - 0.55 }
⇕
(- ∞, - 5.45) ⋃ (- 0.55, ∞ )
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around - 5.45 and - 0.55. 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= 1 and b= 6 into - b2a.
