Book 1 - Year 11 Flashcards
Complete the square
x² - 2x - 16
(x - 1) ² - 17
Complete the square
3x² + 2x - 4
3(x + 1/3)² - 13/3
What is the name of the u or n shape on a quadratic graph?
Parabola
Find the turning point of
-2x² + 4x - 1
( 1, -1)
[the completed square -2(x - 1)² + 1]
Solve as a surd
x² - 10x - 5 = 0
x = 5 +/-√30
Solve to 2 dp
x² + 2x - 9 = 0
x = 2.16 x = -4.16
What is the discriminant?
b² - 4ac
the section under the √ in the quadratic formula
How many roots does a positive discriminant suggest?
2 solutions/ roots
How many roots does a negative discriminant suggest?
0 solutions/ roots
How many roots does a discriminant that equals 0 suggest?
1 solutions/ roots
Find the discriminant of the equation x² + 3x + 5 = 0 and explain what it tells you
0=[(3)²-4(1x5)]
=-11
it is a negative discriminant meaning that there are no real solutions
Find the discriminant of the equation 25x² - 30x + 9 = 0 and explain what it tells you
0=[(-30)²-4(25x9)]
=0
the discriminant =0 meaning there is 1 possible solution
Find the discriminant of the equation 3x² + 2x - 4 = 0 and explain what it tells you
0=[(2)²-4(3x-4)]
=52it is a positive discriminant meaning there are 2 possible solutions
Show that x² - 12x + 40 > 0 for all real values of x
= 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
[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?
(Draw the graph)
The graph doesn’t cross the x axis and therefore there are no real solutions
Find the values of k for which 2x² + kx + 25/8 = 0 has real roots
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
Find the values of k for which 2x² + kx + 25/8 = 0 has no real roots
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
What is the fastest way to find the x-coordinate of the turning point if you have the roots?
Find the midpoint of the roots
What are the other names for turning point?
Stationary point
Maximum/ Minimum point
Find the y-intercept, roots and turning point of the graph y = x² + 8x - 65 by completing the square
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)
Find the y-intercept, roots and turning point of the graph y = x² + 10x + 16 by factorising
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
Describe how the intersection method works to solve quadratics?
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
How would you complete the intersection method for x² -9x - 10 = 0 if you have a graph of y = x² - x - 10
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
