Maths Flashcards
Solve the inequality x^2 - 2x - 3 >= 0
x^2 - 2x - 3 = (x + 1)(x - 3)
Line crosses x axis at x values of -1 and 3
The line is above or equal to the x axis when x <= -1 or x >= 3 so this is the inequality
Complete the square of x^2 - 8x + 3 = 0
x^2 - 8x = -3
(8 / 2 = 4 and 4^2 = 16)
x^2 - 8x + 16 = -3 + 16
(x - 4)^2 = 13
x - 4 = +- sqr(13)
Calculate answers from here
How do you complete the square?
In terms of ax^2 + bx + c
Move c to the other side of the equation
Take b, halve it, square the answer and add the result to both sides of the equation
Factorise the left side
Square root both sides and put a plus / minus sign before the right side
Calculate x from here
How do you solve quadratic inequalities?
For x^2 - 2x - 3 < 0
(x + 1)(x - 3) < 0
So the curve crosses the x axis when x = -1 and x = 3
Because we are looking for <0, the inequality is -1 < x < 3 because the line is under the x axis between x values of -1 and 3
What else is completing the square useful for?
Identifying the line of symmetry and the vertex
For example,
y = x^2 - 8x + 3 can be written as y = (x - 4)^2 - 13
From this, the line of symmetry is x = 4 and the vertex is (4,-13)
How do you complete the square?
Revise from book
What is the bowtie theorem?
If every point of a bowtie shape touches the circumference, the two angles are equal and the two angles at the bottom are equal
What is the theorem that tells us about tangents meeting at an external point?
2 tangents meeting at an external point are equal in length
What is a cyclic quadrilateral?
A quadrilateral where every point is on the circumference of the circle
Using the same shape as the last question what relates POQ and PXQ?
POQ = 180 - PXQ
What is the theorem linking a tangent and a radius?
The angle between a tangent and a radius is 90 degrees
What are supplementary angles?
C angles
The 2 angles equal 180 degrees
What is the arrowhead theorem?
The angle at the centre is twice the angle at the circumference
What is the theorem that tells us when a line will bisect a chord?
The perpendicular from the centre to a chord bisects the chord
What is the cyclic quadrilateral theorem?
Opposite angles in a cyclic quadrilateral add up to 180 degrees
How do you calculate the interior angle size of a regular polygon?
(180 * (number of sides - 2)) / number of sides
What are corresponding angles?
F angles
The two angles equal each other
What are alternate angles?
Z angles
The two angles equal each other
How do you calculate the external angle size of a regular polygon?
360 / number of sides
What is alternate segment theorem?
If a triangle always touching the circumference is labelled ABC and there is a tangent (T) at point B, the angle between T and BC = BAC and the angle between T and BA = ACB
If the centre is O, and there are 2 tangents called P and Q that meet at point X and a line is drawn from X to O, which angles are the same?
OPX and OQX
PXO and QXO
When drawing a graph that has negative and positive values, what do you have to remember?
Draw the graph with a cross axis
What do you have to remember in probability tree questions?
Make a section of the tree for each named condition
In an inequality graph, what do dotted and solid lines mean?
Dotted line is <>
Solid line is <= and >=
In an inequality diagram, what do hollow and filled dots mean?
Hollow is <>
Full is <= and >=
