Equations

Question 1

What is the expansion of a perfect square?

(a + b)² =

(a - b)² =

Answer 1

Square of each term plus twice the product

(a + b)² = a² + 2ab + b²

(a - b)² = a² -2ab + b²

Question 2

what is the expansion of the difference of two squares?

(a + b)(a - b) =

Answer 2

The square of the first minus the square of the second.

(a + b)(a -b) = a² - b²

Question 3

Factorise the diffence of two squares

a² - b² =

eg factorisae x²- 9

Answer 3

(a - b)(a + b)

eg. (x + 3)(x - 3)

Question 4

Factorisng expression with no coefficient infront of the x²

eg x² + 5x + 6

Answer 4

Think

rule: Find two numbers that add to give the middle turn and have a product of the last term

1. find two numbers that add to give 5 and have a product of 6 ie 2,3

x² + 5x + 6= (x +2)(x +3)

Question 5

Rules with the signs when factorising

1. if all positive then both brackets will be positive
2. if the product is positve and the sum is negative then both brackets are negative.
3. if the product is negative then the sum could either be positive or negative and the brackets have one of each

Answer 5

examples

1. x² + 8x + 7 = (x + 7)(x + 1) both pos
2. x²- 8x + 15 = (x -5)(x - 3) both neg
3. x² +4x -21 = (x + 7)(x - 3) one of each

Question 6

Fractorising an expression with a coeeficient in front of the x²

eg. 5x² +13x - 6

Answer 6

5 x -6 = -30

Think: we need two numbers with a product of -30 and a sum of +13, that is 15 and -2

Apply: Place the coefficient of x² together with x at the beginning of each bracket and divide the whole expression by this coefficient.

(5x + 15)(5x -2)

⁵

^{(x + 3)(5x -2)}

Question 7

What is special about the following set of numbers?

1, 8, 27, 64, 125

Answer 7

They are all perfect cubes.

Question 8

What is special about the following set of numbers.

1, 4, 9, 16, 25, 36, 49, 64, 81…

Answer 8

They are all perfect squares!