Exercise 27 Page 211

We will start by rearranging the given quadratic inequality. 0 > - 2x^2+4x+4 ⇔ - 2x^2+4x+4 < 0 We will now sketch the related quadratic function. To do so, we first need to identify the values of a, b, and c. y= - 2x^2 + 4x + 4 We see that a= - 2, b= 4, and c= 4. Let's substitute these values into the Quadratic Formula to find the roots of - 2x^2+4x+4=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, - 2x^2+4x+4 <0, consists of x-values for which the graph of the related quadratic function lies below the x-axis. The graph opens downward, since a= - 2 is less than zero.

We see that the graph lies below the x-axis at about x< - 0.73 and x > 2.73. { x| x < - 0.73 or x > 2.73 } ⇕ (- ∞, - 0.73) ⋃ (2.73, ∞ )

Showing Our Work

Drawing the parabola using roots and vertex

We already know the roots, which are around - 0.73 and 2.73. 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.

- b/2a

SubstituteII

a= - 2, b= 4

- 4/2( - 2)

MultPosNeg

a(- b)=- a * b

- 4/- 4

RemoveNegFracAndDenom

- a/- b= a/b

4/4

CalcQuot

Calculate quotient

1

Finally, to find the y-coordinate of the vertex, we will substitute 1 for x in the related function y=- 2x^2+4x+4.

y=- 2x^2+4x+3

Substitute

x= 1

y=- 2( 1)^2+4 ( 1) +4

▼

Simplify right-hand side

CalcPow

Calculate power

y=- 2(1)+4(1) +4

Multiply

Multiply

y=- 2+4+4

AddTerms

Add terms

y=6

The vertex is (1,6). This point, along with the roots, is helpful to graph a parabola.