algebra 1

Question 1

proof

Answer 1

logical argument for a mathematical statement. Shows that something must be either true or false

Question 2

direct proof

Answer 2

• assume that a statement, P, is true
• use P to show that another statment, Q, is true

Question 3

proof by exhaustion

Answer 3

• list a set of cases that exhausts all possibilities
• show that the statement is true in each and every case

Question 4

index laws

Answer 4

• any number raised to the power zero is 1
• negative powers may be written as reciprocals eg x^-n= 1/x^n
  -any base raised to the power of a unit fraction is a root eg x^1/n= 1/rootx
  -when multiplying terms you add the indices
• to divide terms you subtract the indicies
• to raise one term to another power you multiply the indicies

Question 5

what is a rational number

Answer 5

one that you can write exactly in the form p/q where p and q are integers and q isn’t 0

Question 6

surd laws

Answer 6

√a x √b = √ab
√a /√b= √ a/b

you can also use the difference of two squares to simplify a fraction with a surd at the bottom to make it more rational and put the surd on the top

Question 7

completing the square

Answer 7

ax^2 + bx +c = a(x+b/2a) ^2 +q

Question 8

quadratic formula

Answer 8

x= −b±√b2−4ac
————–
2a

Question 9

discriminant

Answer 9

b^2- 4ac

if discriminant >0 the quadratic has 2 distinct roots and the curve crosses the x-axis at two distinct points
if discriminant =0 the quadratic has one repeated root and the x-axis is a tangent to the curve at this point

if discriminant <0 the quadratic has no real roots and the curve doesnt cross the x-axis at any point

Question 10

ways of solving simultaneous equations

Answer 10

• graphically
  -by eliminating one of the variables
• substitution

Question 11

ways of writing equation of a straight line

Answer 11

y-y1= m(x-x1)
where (x1, y1)is a point on the line

Question 12

gradient of a straight line

Answer 12

y2-y1
……………….
x2-x1

Question 13

distance between two points

Answer 13

d=√((x_2-x_1)²+(y_2-y_1)²

Question 14

midpoint of the line

Answer 14

x_1 +x_2 , y_1 +y_2
———— ————-
2 2

Question 15

equation of circle

Answer 15

(x-a) ^2 + (y-b)^ 2 = r^2

Question 16

circle theorems

Answer 16

• if a triangle passes through the centre of the circle, and all three corners touch the circumference of the circle, then the triangle is right- angles
• the perpendicular line from the centre of the circle to a chord bisects the chord
• any tangent to a circle is perpendicular to the radius at the point of contact (B)

Question 17

ways of representing inequalities

Answer 17

you can represent inequalities on a number line.
You use a coloured/ shaded circle when representing greater or equal to, or less than or equal to
you use an empty circle when representing > or <
- interval notation
- set notation
when you divide or multiply by negative number inequality sign flips

Question 18

set notation

Answer 18

uses curly brackets and the intersection and union symbols
eg -1< x ≤ 4
can be written as:
{x: x> -1 } ∩ {x : x ≤ 4 }
can be simplified to
{ x: -1 <x ≤4}

Question 19

interval notation

Answer 19

uses square and round brackets
[,] for weak inequalities of ≤ or ≥
(,) for < or >
infinity or negative infinity can be used with round brackets to show the upper or lower bound
- union symbol can be used to show two separate inequalities
Interval notation uses different brackets to indicate whether a number is included or not
- [ or ] mean included
- ( or ) mean excluded
- (4,8] means 4 < x < 8
- Note ∞ always uses ( or )