Find the roots and use them to graph the related function.
{ x| x < - 0.73 or x > 2.73 }
Practice makes perfect
We will start by rearranging the given quadratic inequality.
0 > - 2x^2+4x+4 ⇔ - 2x^2+4x+4 < 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= - 2x^2 + 4x + 4
We see that a= - 2, b= 4, and c= 4. 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=- 4±sqrt(48)/- 4
x=- 4 + sqrt(48)/- 4
x=- 4 - sqrt(48)/- 4
x=- 4/- 4+sqrt(48)/- 4
x=- 4/- 4-sqrt(48)/- 4
x=1-sqrt(48)/4
x=1+sqrt(48)/4
x≈ - 0.73
x≈ 2.73
The solution of the given quadratic inequality, - 2x^2+4x+4 <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= - 2 is less than zero.
We see that the graph lies below the x-axis at about x< - 0.73 and x > 2.73.
{ x| x < - 0.73 or x > 2.73 }
⇕
(- ∞, - 0.73) ⋃ (2.73, ∞ )
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around - 0.73 and 2.73. 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= 4 into - b2a.
