Exercise 19 Page 211

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.

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.

- b/2a

SubstituteII

a= 1, b= - 9

- - 9/2( 1)

IdPropMult

Identity Property of Multiplication

- - 9/2

RemoveNegFracAndNum

- - a/b= a/b

9/2

Finally, to find the y-coordinate of the vertex, we will substitute 92 for x in the related function y=x^2-9x+9.

y=x^2-9x+9

Substitute

x= 9/2

y=( 9/2 )^2-9 ( 9/2 ) +9

▼

Simplify right-hand side

CalcPow

Calculate power

y=81/4-9 ( 9/2 ) +9

MoveLeftFacToNum

a*b/c= a* b/c

y=81/4 - 81/2+9

ExpandFrac

a/b=a * 2/b * 2

y=81/4 - 162/4+9

WriteAsFrac

Write as a fraction

y=81/4 - 162/4+ 36/4

AddSubFrac

Add and subtract fractions

y=- 45/4

MoveNegNumToFrac

Put minus sign in front of fraction

y=- 45/4

The vertex is ( 92,- 454 ). This point, along with the roots, is helpful to graph a parabola.