Find the roots and use them to graph the related function.
{ x| x < 1 or x > 4 }
Practice makes perfect
We will start by rearranging the inequality.
0 < x^2-5x+4 ⇔ x^2-5x+4 > 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=x^2-5x+4 ⇔ y= 1x^2+( - 5)x+ 4
We see that a= 1, b= - 5, and c= 4. Let's substitute these values into the Quadratic Formula to find the roots of x^2-5x+4=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=5 ± 3/2
x=5 + 3/2
x=5 - 3/2
x=8/2
x=2/2
x=4
x=1
The solution of the given quadratic inequality, x^2-5x+4>0, consists of x-values for which the graph of the related quadratic function lies above the x-axis.The graph opens upwards since a= 1 is greater than zero.
We see that the graph lies above the x-axis at x<1 and x >4.
{ x| x < 1 or x > 4 }
⇕
(- ∞, 1) ⋃ (4, ∞ )
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are 1 and 4. 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= - 5 into - b2a.
