Find the roots and use them to graph the related function.
{ x| x < - 1.42 or x > 8.42 }
Practice makes perfect
We will start by rearranging the given quadratic inequality.
0 > - x^2+7x+12 ⇔ - x^2+7x+12 < 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= - 1x^2 + 7x + 12
We see that a= - 1, b= 7, and c= 12. Let's substitute these values into the Quadratic Formula to find the roots of - x^2+7x+12=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=- 7±sqrt(97)/- 2
x=- 7 + sqrt(97)/- 2
x=- 7 - sqrt(97)/- 2
x=7/2-sqrt(97)/2
x=7/2+sqrt(97)/2
x≈ - 1.42
x≈ 8.42
The solution of the given quadratic inequality, - x^2+7x+12 < 0, consists of x-values for which the graph of the related quadratic function lies below the x-axis. The graph opens downward, since a= - 1 is less than zero.
We see that the graph lies below the x-axis at about x < - 1.42 and x > 8.42.
{ x| x < - 1.42 or x > 8.42 }
⇕
(- ∞, - 1.42) ⋃ (8.42, ∞ )
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around - 1.42 and 8.42. 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= 7 into - b2a.
