Algebra

Question 1

What is the Quadratic Formula?

Answer 1

Question 2

What is the general equation for Factorising the Difference of Two Squares (D.O.T.S.)?

Answer 2

x²-y² = (x-y) (x+y)

= x² + xy - xy - y²

= x² - y²

Question 3

What is the general equation for Factorising the Difference of Cubes?

Answer 3

x³ - y³ = (x - y) (x² + xy + y²)

Question 4

What is the general equation for Factorising the Sum of Cubes?

Answer 4

x³ + y³ = (x + y) (x² - xy + y²)

Question 5

How can you Factorise the Sum of Two Squares?

Hint: involves Imaginary numbers.

Answer 5

x² + y² = x² - (-y²)

= x² - (i²y²)

= x² - (iy)²

= (x + iy) (x - iy)

Question 6

What is the algebraic expansion of (a + b)²?

Answer 6

(a + b)² = a² + 2ab + b²

Question 7

What is the algebraic expansion of (a - b)²?

Answer 7

(a - b)² = a² - 2ab + b²

Question 8

Simplify a^maⁿ

Answer 8

a^maⁿ = a^m+n

Question 9

Simplify a^m/aⁿ

Answer 9

a^m/aⁿ = a^m-n

Question 10

Simplify (aⁿ)^m

Answer 10

(aⁿ)^m = a^nm

Question 11

Simplify a⁰

Answer 11

a⁰ = 1

Question 12

Simplify a^1/n

Answer 12

a^1/n = ⁿ√a

Question 13

Simplify a^m/n

Answer 13

a^m/n = ⁿ√a^m

Question 14

The log of a product is said to be equal to the sum of two logs.

Show what this means with relation to log_a(xy)

Answer 14

log_a (xy) = log_a x + log_a y

Question 15

The log of a quotient is said to be equal to the difference of two logs to the same base.

Show what this means with relation to log_a (x/y)

Answer 15

log_a (x/y) = log_a x - log_a y

Question 16

Simplify log_a xⁿ

Answer 16

log_a xⁿ = n•log_a x

Question 17

The rules of logs have a useful result when related to problems such as a^{loga of x}

what is it?

Answer 17

a^{loga of x} = x

inversely

x = a^{loga of x}

Question 18

The rules of logs have a number of obscure useful properties.

How can they be used to simplify the following?

a^xloga^y

Answer 18

a^xloga^y = y^x

Question 19

It is possible to change the base of a logarithm.

Show how using log_a x

Answer 19

log_a x = log_b x / log_b a

Question 20

What does (a + b)³ expand to?

Answer 20

(a +b)³ = a³ + 3a²b + 3ab² + b³

Question 21

What does (a - b)³ expand to?

Answer 21

(a - b)³ = a³ - 3a²b + 3ab² - b³