Exercise 20 Page 286

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^2-2x-24 ⇔ y= 1x^2+( - 2)x+ - 24 We see that a= 1, b= - 2, and c= - 24. Let's substitute these values into the Quadratic Formula to find the roots of x^2-2x-24=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^2-2x-24≤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= 1 is greater than zero.

We see that the graph lies on and below the x-axis between x=- 4 and x=6. { x| - 4 ≤ x ≤ 6 } ⇕ [- 4,6 ]

Showing Our Work

Drawing the parabola using roots and vertex

We already know the roots, which are around - 4 and 6. 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= - 2 into - b2a.

- b/2a

SubstituteII

a= 1, b= - 2

- - 2/2( 1)

IdPropMult

Identity Property of Multiplication

- - 2/2

RemoveNegFracAndNum

- - a/b= a/b

2/2

CalcQuot

Calculate quotient

1

Finally, to find the y-coordinate of the vertex, we will substitute 1 for x in the related function y=x^2-2x-24.

y=x^2-2x-24

Substitute

x= 1

y=( 1)^2-2 ( 1) -24

▼

Simplify right-hand side

BaseOne

1^a=1

y=1-2(1) -24

IdPropMult

Identity Property of Multiplication

y=1-2 -24

AddSubTerms

Add and subtract terms

y=- 25

The vertex is ( 1,- 25). This point, along with the roots, is helpful to graph a parabola.