Book 1 - Year 11

Question 1

Complete the square

x² - 2x - 16

Answer 1

(x - 1) ² - 17

Question 2

Complete the square

3x² + 2x - 4

Answer 2

3(x + 1/3)² - 13/3

Question 3

What is the name of the u or n shape on a quadratic graph?

Answer 3

Parabola

Question 4

Find the turning point of

-2x² + 4x - 1

Answer 4

( 1, -1)

[the completed square -2(x - 1)² + 1]

Question 5

Solve as a surd

x² - 10x - 5 = 0

Answer 5

x = 5 +/-√30

Question 6

Solve to 2 dp

x² + 2x - 9 = 0

Answer 6

x = 2.16
x = -4.16

Question 7

What is the discriminant?

Answer 7

b² - 4ac

the section under the √ in the quadratic formula

Question 8

How many roots does a positive discriminant suggest?

Answer 8

2 solutions/ roots

Question 9

How many roots does a negative discriminant suggest?

Answer 9

0 solutions/ roots

Question 10

How many roots does a discriminant that equals 0 suggest?

Answer 10

1 solutions/ roots

Question 11

Find the discriminant of the equation x² + 3x + 5 = 0 and explain what it tells you

Answer 11

0=[(3)²-4(1x5)]
=-11
it is a negative discriminant meaning that there are no real solutions

Question 12

Find the discriminant of the equation 25x² - 30x + 9 = 0 and explain what it tells you

Answer 12

0=[(-30)²-4(25x9)]
=0
the discriminant =0 meaning there is 1 possible solution

Question 13

Find the discriminant of the equation 3x² + 2x - 4 = 0 and explain what it tells you

Answer 13

0=[(2)²-4(3x-4)]

=52it is a positive discriminant meaning there are 2 possible solutions

Question 14

Show that x² - 12x + 40 > 0 for all real values of x

Answer 14

= x² - 12x + 40
= (x - 6)² + 4 
(x - 6)² ≥ 0
Therefore (x - 6)² + 4 ≥ 4
Therefore x² - 12x + 40 > 0 for all real values of x QED

Question 15

[Show that x² - 12x + 40 > 0 for all real values of x
(x - 6)² + 4
(x - 6)² ≥ 0
x - 6)² + 4 ≥ 4]
What does this tell you about the graph of y = x² - 12x + 40?

Answer 15

(Draw the graph)

The graph doesn’t cross the x axis and therefore there are no real solutions

Question 16

Find the values of k for which 2x² + kx + 25/8 = 0 has real roots

Answer 16

2x² + kx + 25/8 = 0 
16x² + 8kx + 25 = 0 
a = 16
b = 8k
c = 25
(8k)² - 4(16 x 25) ≥ 0
64k² - 1600 ≥ 0
(k + 5) (k - 5)  ≥ 0
Draw a graph of the discriminant (with roots 5 and negative 5)
ANSWER:
k ≥ 5
k ≤ -5

Question 17

Find the values of k for which 2x² + kx + 25/8 = 0 has no real roots

Answer 17

2x² + kx + 25/8 = 0 
16x² + 8kx + 25 = 0 
a = 16
b = 8k
c = 25
(8k)² - 4(16 x 25) < 0
64k² - 1600 < 0
(k + 5) (k - 5)  < 0
Draw a graph of the discriminant (with roots 5 and negative 5)
ANSWER:
-5 < k < 5

Question 18

What is the fastest way to find the x-coordinate of the turning point if you have the roots?

Answer 18

Find the midpoint of the roots

Question 19

What are the other names for turning point?

Answer 19

Stationary point

Maximum/ Minimum point

Question 20

Find the y-intercept, roots and turning point of the graph y = x² + 8x - 65 by completing the square

Answer 20

y = (x + 4)² - 16 - 65
y = (x + 4)² - 81
tp: (- 4, -81)
0 = (x + 4)² - 81
81 = (x + 4)² 
\+/-9 = x + 4
- 4 +/- 9 = x
x = 5
x = -13
roots = ( 5, 0) and ( -13, 0)
y intercept = ( 0, -65)

Question 21

Find the y-intercept, roots and turning point of the graph y = x² + 10x + 16 by factorising

Answer 21

y = x² + 10x + 16 
y = (x + 8) (x + 2)
x = -8
x = -2
roots = ( -8, 0) and ( -2, 0)
(-8 + -2) ÷ 2= -5
(tp:(-5, y))
y = (-5)² + 10(-5) + 16 
y = 25 - 50 + 16
y = -9
tp: ( -5, -9)
y intercept = 16

Question 22

Describe how the intersection method works to solve quadratics?

Answer 22

You find the difference between the graph they have drawn and the one you need to solve and draw the line of the difference
(This is where the x axis would be on your graph)
The intersects are the solutions

Question 23

How would you complete the intersection method for x² -9x - 10 = 0 if you have a graph of y = x² - x - 10

Answer 23

y = x² - 8x - 10
0 = x² -9x - 10 (subtract these from each other)
__________
y = x
you would draw the line y = x onto the graph they have given
where both lines intersect is the solution