Find the roots and use them to graph the related function.
{ x| - 2.3 < x < 1.3 }
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=- 3 x^2-3x+9 ⇔ y= - 3x^2+( - 3)x+ 9
We see that a= - 3, b= - 3, and c= 9. Let's substitute these values into the Quadratic Formula to find the roots of - 3x^2-3x+9=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=3±sqrt(117)/- 6
x=3 + sqrt(117)/- 6
x=3 - sqrt(117)/- 6
x=3/- 6+ sqrt(117)/- 6
x=3/- 6- sqrt(117)/- 6
x=- 3/6- sqrt(117)/6
x=- 3/6+ sqrt(117)/6
x≈ - 2.3
x≈ 1.3
The solution of the given quadratic inequality, -3x^2-3x+9 > 0, consists of x-values for which the graph of the related quadratic function lies above the x-axis. The graph opens downward, since a= - 3 is less than zero.
We see that the graph lies above the x-axis between x=- 2.3 and x=1.3.
{ x| - 2.3 < x < 1.3 }
⇕
(- 2.3,1.3 )
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around - 2.3 and 1.3. 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= - 3 and b= - 3 into - b2a.
