Section 1.1

Question 1

How do you rationalise these fractions:
1/√a
1/a+√b
1/a-√b

Answer 1

1) Multiply numerator & denominator by √a
2) Multiply num & den by a - √b
3) Multiply num & den by a + √b

Question 2

How would you solve this question? (Include name of method)

x-y = 7
y^2+xy+2x = 5

Answer 2

Substitution Method

x = 7+y
y^2 + y(7+y) + 2(7+y) = 5
2y^2 + 9y + 9 = 0

Then solve quadratic, substituting the y-values back into one equation to get the corresponding x-values

Question 3

For each discriminant, state the number of solutions for a pair of simultaneous equations:

b^2 - 4ac > 0

b^2 - 4ac = 0

b^2 - 4ac < 0

Answer 3

1) Two real solutions
2) 1 real solution (or 2 of the same)
3) No real solutions

Question 4

What is the formula for completing the square?

Answer 4

a(x+b/2a)^2 + (c-b^2/4a)

Question 5

When sketching a graph, what can you use to identify how many roots are there?

Answer 5

Use the discriminant, following the rules in order to identify the amount

Question 6

How would you solve double-ended inequalities?
E.g.
32 < 5x+7 < 57

Answer 6

Split it into 2 smaller inequalities
E.g.

32 < 5x+7 & 5x+7 < 57

Question 7

How would you find a set of values which satisfies both inequalities?
E.g.
4(x+9) < 9x + 56
3x - 8 < 12 - x

Answer 7

1st) Solve separately
x > -4 , x < 5

So set of values would be:

-4 < x < 5

Question 8

For Quadratic Inequalities, what values do we look for when: E.g. eqn x^2+3x-2

1) x^2+3x-2 > 0
2) x^2+3x-2 < 0

Answer 8

1) Those above the x-axis/positive y-axis (or number in place of 0)
2) Below x-axis, negative y-axis

Question 9

When given Quadratic Inequality with equations on either side, how do you work out desired values?

E.g.
x^2+5x+1 < 5x+5

Answer 9

Move all onto 1 side, so:
x^2 - 4 < 0 
Then work out as normal,
x^2 < 4
x < 2, x > -2

Remember that it looks for values less than 0 so under x-axis, mean inequality will be:
-2 < x < 2

Question 10

Practice Long Division for Factorising Polynomials:

E.g. x^3 - 5x^2 - 6x - 56 by (x-7)

Answer 10

Useful for if asked in question - write down method
Answer: x^2 + 2x + 8

Further Practice - P. 141