Simplifying and manipulating expressions

In the diagram above, the expression in each block in the top two rows is found by using the following rule:

The expression in any block is the sum of the expressions in the two blocks below it.

Some expressions are already shown. Use the rule to fill in the empty blocks in the diagram.

You must simplify your answers.

In the diagram above, the expression in each block in the top two rows is found by using the following rule:

The expression in any block is the sum of the expressions in the two blocks below it.

Some expressions are already shown. Use the rule to fill in the empty blocks in the diagram.

You must simplify your answers.

Express − as a single fraction in its simplest form.

Factorise 2x² + 3x − 20.

Hence solve the equation 2x² + 3x − 20 = 0.

Solve each of the following equations.

3y − 5 = 19

8 p + 5 = 3p − 25

Factorise w² − 6w.

Use your solutions from part (b) to find the dimensions of the rectangle.

You must justify any decisions that you make.

Prove that (8n +1)²−3 is always an even number, for all integers n.

You must use an algebraic method.

Factorise (2x + 3)² −16.

Simplify 5k − 8k + 6k.

Solve these equations.

15 + x = 60

20 − y = 9

6w = 54

In the diagram shown below, the term in each box in the top two rows is found by using

the following rule.

The term in each box is the sum of the terms in the two boxes below it.

Some terms are already shown.

Use the rule to write down the missing terms in the two empty boxes.

Space for working:

Factorise each of the following.

14a − 35

5x + x²

Solve the following equation.

Factorise x² + 3x − 40, and hence solve x² + 3x − 40 = 0.

Factorise 2x² − 17x + 30.

Make f the subject of the following formula.

Factorise x² + 3x − 40, and hence solve x² + 3x − 40 = 0.

Simplify the expression 5y + 7m − 3y −10m.

Write an expression, in terms of x, to represent each of the following.

5 more than x

x less than 3

half of x

Write down the next two numbers in the following sequence.

Simplify the expression 5y + 7m − 3y −10m.

Factorise x² −8x − 20, and hence solve x² −8x − 20 = 0.

A number machine is shown below.

Calculate the OUTPUT when the INPUT is 1·5.

Write down an expression for the OUTPUT when the INPUT is n.

Factorise x² −8x − 20, and hence solve x² −8x − 20 = 0.

Factorise 6x² +19x +10.

Fully factorise the expression m³ − 25m.

Factorise (p + 7)(p + 29) + 2(p + 7).

The cost of one book is £m.

What is the cost of 15 books in pounds (£)?

Ieuan has 20 oranges. He gives away k oranges.

How many oranges does Ieuan have now?

Solve the equation 3x 10 17.

Simplify 6 f 4g 2 f 9g.

Rhian is n years old.

Samir is 7 years younger than Rhian. Nigel is twice as old as Samir.

Write down an expression, in terms of n, for Nigel’s age.

Solve the following equations.

3x 10 17

Simplify 6 f 4g 2 f 9g.

Rhian is n years old.

Samir is 7 years younger than Rhian. Nigel is twice as old as Samir.

Write down an expression, in terms of n, for Nigel’s age.

Simplify each of the following. Circle your answer in each case.

m⁴ m³

Write down an expression for the nth term of the following sequence.

List all of the integers that satisfy the following inequality.

13 < 2 n < 19

Solve the equation 7 5x 2 3x 8.

Make f the subject of the formula h 13 2 f.

Factorise 15x 35y.

Simplify each of the following. Circle your answer in each case.

m⁴ m³

Write down an expression for the nth term of the following sequence.

List all of the integers that satisfy the following inequality.

13 < 2 n < 19

Show that 16x² 8x 3 0.

Solve the equation 16x² 8x 3 0.

You must use an algebraic method.

Find the length of BC.

You must justify any decision that you make.

Solve the equation 7 5x 2 3x 8.

Make f the subject of the formula h 13 2 f.

Factorise 15x 35y.

Factorise 8x² 18.

Hence solve 8x² 18 0.

Hence, sketch the graph of y 8x² 18 on the axes below.

Mark clearly the coordinates of any point where this graph crosses an axis.

Space for working:

Use the quadratic formula to solve the equation 1.

Give your answers correct to 2 decimal places.

You must show all your working.

Solve the equation 7p 3 60.

Simplify the expression 6a 7b 2a 8b.

Simplify 3a 2a a.

How many dots will there be in Diagram 6?

Solve the equation 7p 3 60.

Simplify the expression 6a 7b 2a 8b.

Simplify the following expressions.

7a(a + 5) − 2(3a² + 6a − 7)

Write 0·034 35 correct to two significant figures.

Circle your answer.

0·03 0·033 0·0344 0·034 0·034 00

Convert 6∙7 m² into cm².

Circle your answer.

670 6700 67 000 670 000 6 700 000

Factorise 12e 15.

Circle your answer.

27e 34e 5 12e 15 5(12e + 3) 150 8e 3

The rectangle below has: • a length of 3y • a width of y + a • an area of 3y² +12y.

Find the value of a.

You must show all your working.

Another rectangle has a width of 4x 10.

Given that x is a whole number, explain why the value of x cannot be less than 3.

The perimeter of the rectangle is 14x 4.

Find the length of the rectangle in terms of x.

Simplify the following expressions.

7a(a + 5) − 2(3a² + 6a − 7)

A rectangle has a width of 4x 10.

The perimeter of the rectangle is 14x 4.

Find the length of the rectangle in terms of x.

Expand and simplify 2h 3t5h 7t.

Fully factorise the expression k³p – kp³.

Simplify 12a 19a 10a.

Solve the following equations.

Describe in words the rule for continuing the sequence below.

79, 65, 51, 37,... Rule:

Write down the next term in the sequence below.

Adrian has n grapes. He eats 4 of them.

Write down, in terms of n, the total number of grapes Adrian now has.

Expand 43p 5.

Make m the subject of the formula w 8m 3.

Expand and simplify y 5y 4.

Factorise x² 13x 40.

Explain how you can check that your answer to part (i) is correct.

Marian is competing in a race.

The race is (6x + 5) miles long. She completes the race in x hours. Her average speed during the race is (2x + 3) miles per hour.

Calculate how long Marian takes to complete the race.

You must use an algebraic method (not trial and improvement).

Make m the subject of the formula w 8m 3.

Expand and simplify y 5y 4.

Factorise 8x² + 6xy.

Factorise x² + 13x + 40.

Explain how you can check that your answer to part (i) is correct.

Show that 5x³ 15x 132.

A solution of the equation

lies between 2 and 3. Use the method of trial and improvement to find this solution correct to 1 decimal place.

You must show all your working.

Hence, find the height of the cuboid.

Express the following as a single fraction in its simplest form.

Shape A is a square and Shape B is a rectangle, as shown below.

The area of square A is equal to the area of rectangle B.

Form an equation in x and hence calculate the value of x. Give your answer correct to 1 decimal place.

You must show all your working.

Write the next term in the sequence below.

Describe the rule for continuing the following sequence.

77, 64, 51, 38, 25,...

Rule:

A dog is x years old. Another dog is 2 years younger.

Write down, in terms of x, the age of the younger dog.

A rectangle has sides of length 2(3 a – 7) cm and (5 a + 4) cm.

Form an expression, in terms of a, for the perimeter of this rectangle.

You must simplify your expression.

Expand x(x² + 7).

Expand and simplify (x – 5)(3x − 4).

Sarah buys and sells antique clocks. On Monday, Sarah had n clocks. At the end of the day on Tuesday, she had 5 times as many clocks as she had on Monday. On Wednesday, she sold 27 clocks.

At the end of the day on Wednesday, Sarah had fewer clocks than she had on Monday.

Write an inequality, in terms of n, that shows this information.

Solve your inequality to find the greatest number of clocks that Sarah could have

had on the Monday.

Factorise x² − 4x − 12, and hence solve x² − 4x − 12 = 0.

A rectangle has sides of length 2(3 a – 7) cm and (5 a + 4) cm.

Form an expression, in terms of a, for the perimeter of this rectangle.

You must simplify your expression.

Expand x(x² + 7).

Expand and simplify (x – 5)(3x − 4).

Sarah buys and sells antique clocks. On Monday, Sarah had n clocks. At the end of the day on Tuesday, she had 5 times as many clocks as she had on Monday. On Wednesday, she sold 27 clocks.

At the end of the day on Wednesday, Sarah had fewer clocks than she had on Monday.

Write an inequality, in terms of n, that shows this information.

Solve your inequality to find the greatest number of clocks that Sarah could have

had on the Monday.

Factorise x² − 4x − 12, and hence solve x² − 4x − 12 = 0.

Simplify the following expression.

Show that

Solve the equation

You must use an algebraic method and show all your working.

Give your answers correct to 2 decimal places.

Evaluate the length of AC.

You must justify any decision that you make.

Simplify the expression 9 g – 5 g + 12 g.

Solve the equation 5y = 45.

Solve the equation w – 16 = 14.

Solve the equation 4x + 7 = 10.

Solve 5(2 x + 3) = 20.

A number machine is shown below.

Write down an expression for the OUTPUT when the INPUT is n.

Solve the equation 4 x + 7 = 10.

Simplify 8d – 6e – 3d + 4e.

Factorise x² – 7x + 12, and hence solve x² – 7x + 12 = 0.

Expand and simplify (5x – 2)².

Solve 5(2 x + 3) = 20.

Factorise 7a + 21.

A number machine is shown below.

Write down an expression for the OUTPUT when the INPUT is n.

Factorise x² – 7x + 12, and hence solve x² – 7x + 12 = 0.

Expand and simplify (5x – 2)².

Express the following as a single fraction in its simplest form.

Factorise 4x² – 81.

Factorise 7x² + 10x – 8.

Factorise (x + 2)³ + 5(x + 2)².

Use the quadratic formula to solve (5 x + 3)(5 x – 3) = 19 x.

Give your answers correct to 2 decimal places.

You must show all your working.

Solve these equations.

7x = 56

23 – x = 9

Sian has n boxes. Each box contains 8 pens.

How many pens does Sian have altogether?

Number of pens is_{………........ ………………}

Meic had m CDs. He gave 3 CDs to a friend.

How many CDs does Meic have now?

Number of CDs is_{………........ ………………}

Calculate the value of 3 x + 4 y when x = −6 and y = 5.

Simplify the expression 9g – 4f – 3g – 5f.

Expand 4(3x – 5).

Calculate the value of 3 x + 4 y when x = −6 and y = 5.

Simplify the expression 9g – 4f – 3g – 5f.

Solve the equation 3m − 7 = 8.

Expand 4(3x – 5).

Factorise x² + 4x – 21. Hence, solve x² + 4x – 21 = 0.

Factorise x² + 4x – 21. Hence, solve x² + 4x – 21 = 0.

Show that

6 x (8 x + 1) – 3(4 x + 1)(4 x – 1) ≡ 6 x + 3.

Hence, write down the value of x for which 6x(8x + 1) – 3(4x + 1)(4x – 1) = 0.

Simplify the following expression.

Simplify, and then factorise, the following expression.

Show that 2x² + 3x – 50 = 0.

Solve the equation 2x² + 3x – 50 = 0 to calculate the lengths AB and DC.

Give your answers correct to 1 decimal place.

Length of AB =_{……........ ……..} cm Length of DC =_{……........ ……..} cm

A rhombus has an area of 36∙8 cm². The rhombus is enlarged by a scale factor of 7.

Calculate the area of the enlarged rhombus.

Simplify 8 p – 12 p + 9 p.

Solve the following equations.

6x = 48

32 – y = 17

Tom thinks of a number. He multiplies the number by 4. The answer is 76.

What number did Tom think of?

Describe in words the rule for continuing each of the following sequences.

38, 32, 26, 20, …

Rule:

Rule:

Adam has x marbles. His friend gives him 3 more.

Write down, in terms of x, the total number of marbles Adam now has.

Petrol costs £g per litre.

Write down, in terms of g, the cost of 15 litres of petrol.

Factorise 10 a – 15.

Solve the following equations.

13f + 2 = 6f + 5.

n is an integer.

Tick the correct statement below.

You must give an explanation for your decision.

Factorise 7 ab + 11 a.

Factorise x² – 8x.

Expand 4y(2 – 3y).

Write the following expression as a single fraction.

Give your answer in its simplest form.

Solve the following equation.

Factorise 81p² – 1.

Factorise 7t² + 19t – 6.

Use the quadratic formula to solve (3x – 2)² = (x + 1)(x + 2).

Give your answers correct to 2 decimal places.

You must show all your working.

Simplify the expression 15 x – 2 y – 7 x – 4 y.

Solve the equation 2m – 7 = 12.

Calculate the value of 5f + 3g when f = −4 and g = 7.

One of the values below is the solution to k + 3 = 6·9

Circle the correct answer.

2·3 20·7 3·9 9·9 207

Simplify the expression 12w + 15w – w.

Write down the next term in the following sequence.

Describe in words the rule for continuing the following sequence.

4, 12, 36, 108, …

Look at the sequence:

8, 16, 24, 32, …

Will any of the first 100 terms of this sequence be a prime number?

Explain how you know.

Simplify the expression 15 x – 2 y – 7 x – 4 y.

Solve the equation 2m – 7 = 12.

Calculate the value of 5f + 3g when f = −4 and g = 7.

Expand 3x(x² – 2).

Make g the subject of the formula f = 2 – 3g.

Solve 7x – 3 < 29.

What is the greatest integer value of x that satisfies the above inequality?

Factorise the following.

6x – 15

y² + 4y

The nth term of a sequence is given by 3n – 20.

What is the value of the 6th term?

Consider the following statement.

‘There are no odd numbers greater than 50 in this sequence.’

Show that this statement is incorrect.

Expand 3x(x² – 2).

Make g the subject of the formula f = 2 – 3g.

Solve 7x – 3 < 29.

What is the greatest integer value of x that satisfies the above inequality?

Simplify the following expression.

Fully factorise the expression c³ – cd².

Factorise and simplify 5(e – 1)² + 3(e – 1).

Give your answers correct to 2 decimal places.

You must show all your working.

Look at the diagram below.

The term in each square in the top two rows is found by using the following rule:

The term in any square is the sum of the terms in the two squares below it.

Some terms are already shown.

Use the rule to write down the missing terms in the three empty squares.

Space for working:

Solve. 7

Simplify 3f + 7g + f – 4g.

Use the formula 5p + 2q = t to find the value of q when p = 4 and t = 24.6.

Expand and simplify the following expression.

Factorise 12x² + 3xy.

Simplify 3f + 7g + f – 4g.

Use the formula 5p + 2q = t to find the value of q when p = 4 and t = 24.6.

Expand and simplify the following expression.

Factorise 12x² + 3xy.

A ball is thrown upwards from a height of 1 m above the ground.

After t seconds, its height above the ground is h metres, where h is given by

Show that the time taken to reach a height of 4 metres satisfies the equation

5t² – 8t + 3 = 0.

Solve the equation

5t² – 8t + 3 = 0.

Interpret your answer to part (b) in the context of the question.

Factorise 4m² – 289.

Express the following as a single fraction in its simplest form.

Solve the equation (2 x + 5)(3 x – 11) = 7.

Give your answers correct to 2 decimal places.

Write down the next two numbers in the following sequence.

Expand 5(3x – 2).

Solve 9x + 3 = 4x + 5.

Factorise x² – 7x – 18, and hence solve x² – 7x – 18 = 0.

Simplify each of the following and circle the correct answer in each case.

6p⁶ × 3p³

3·4g⁸ ÷ 13·6g²

Factorise x² – 7x – 18, and hence solve x² – 7x – 18 = 0.

Two different squares are constructed. The side length of the smaller square is xcm. The side length of the larger square is 3 cm longer than the side length of the smaller square. The combined area of the two squares is 22∙5 cm².

Show that 4x² + 12x − 27 = 0.

Simplify each of the following and circle the correct answer in each case.

6p⁶ × 3p³

3·4g⁸ ÷ 13·6g²

Show that (10w + 3)(w – 1) – (2 – 3w)² w² + 5w – 7.

Use the quadratic formula to solve the equation w² + 5w – 7 = 0.

Give your answers correct to 2 decimal places.

Solve these equations.

7x = 56

y + 19 = 83

Simplify the expression 12k – 15k + 7k.

Write down the next two numbers in the following sequence.

Find the value of 2x + 7y when x = –3 and y = 10.

Simplify the expression 8k + 3m – 2k – 8m.

The diagram below shows a number machine.

Using the number machine, calculate:

the INPUT when the OUTPUT is 36,

the OUTPUT when the INPUT is n.

Look at the diagram below.

The expression in each circle is found by adding the expressions in the rectangles on either side of the circle. Complete the diagram by writing expressions in the blank circles and the blank rectangle.

You must simplify your expressions.

Factorise x³ − 5x.

Expand and simplify (2x – 3)(x + 4).

Factorise x² − 3x – 28.

The diagram shows two rectangles.

The combined area of both rectangles is 50 cm². By considering the areas of the two rectangles, show that 2x² – 5x – 25 = 0 and hence find the

value of x.

Factorise x³ − 5x.

Expand and simplify (2x – 3)(x + 4).

Factorise x² − 3x – 28.