Find the roots and use them to graph the related function.
{ x| 1.1 < x < 7.9 }
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=x^2-9x+9 ⇔ y= 1x^2+( - 9)x+ 9
We see that a= 1, b= - 9, and c= 9. Let's substitute these values into the Quadratic Formula to find the roots of x^2-9x+9=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=9±sqrt(45)/2
x=9 + sqrt(45)/2
x=9 - sqrt(45)/2
x=9/2+sqrt(45)/2
x=9/2-sqrt(45)/2
x≈ 7.9
x≈ 1.1
The solution of the given quadratic inequality, x^2-9x+9 < 0, consists of x-values for which the graph of the related quadratic function lies below the x-axis. The graph opens upwards since a= 1 is greater than zero.
We see that the graph lies below the x-axis at about 1.1 < x < 7.9.
{ x| 1.1 < x < 7.9 }
⇕
(1.1, 7.9)
Showing Our Work
Drawing the parabola using roots and vertex
We already know the roots, which are around 1.1 and 7.9. 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= - 9 into - b2a.
