Find the roots and use them to graph the related function.
{ x| 0.29 ≤ x ≤ 1.71 }
Practice makes perfect
We will start by rearranging the inequality.
0 ≥ 2x^2-4x+1 ⇔ 2x^2-4x+1 ≤ 0
Now, we are ready to sketch the related quadratic function. To do so, we first need to identify the values of a, b, and c.
y=2x^2-4x+1 ⇔ y= 2x^2+( - 4)x+ 1
We see that a= 2, b= - 4, and c= 1. Let's substitute these values into the Quadratic Formula to find the roots of 2x^2-4x+1=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=4 ± sqrt(8)/4
x=4 + sqrt(8)/4
x=4 -sqrt(8)/4
x=4/4+sqrt(8)/4
x=4/4-sqrt(8)/4
x≈ 1.71
x≈ 0.29
The solution of the given quadratic inequality, 2x^2-4x+1≤0, consists of x-values for which the graph of the related quadratic function lies on and below the x-axis. The graph opens upwards since a= 2 is greater than zero.
We see that the graph lies on and below the x-axis at about x ≥ 0.29 and x ≤ 1.71.
{ x| 0.29 ≤ x ≤ 1.71 }
⇕
[0.29,1.71]
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around 0.29 and 1.71. 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.
