Algebraic notation & vocabulary

For each of the algebraic statements below decide whether it is an equation or an identity.

(x - 1)(x - 2) = 12 is an........

A straight line is parallel to y = 3 x and passes through the point (2, 11).

Find the equation of the line in the form y = mx + c.

For each of the algebraic statements below decide whether it is an equation or an identity.

(x - 1)(x - 2) = 12 is an........

A straight line is parallel to y = 3 x and passes through the point (2, 11).

Find the equation of the line in the form y = mx + c.

A sequence has nth term 2n² + 1.

Prove algebraically that the sum of any two consecutive terms in this sequence is always a

multiple of 4.

In this question, all lengths are in centimetres.

The area of the rectangle is 70 cm².

Show that x² + 5x - 84 = 0.

Solve by factorising.

(b)(i) x =........ or x =........

Find the length of the longer side of the rectangle.

Choose a word from this list which best describes each statement.

x² + 3x + 2 (a)........

(x + 1)(x + 2) = x² + 3x + 2 (b)........

Eight consecutive numbers are written in ascending order in this grid, starting from the top and working left to right.

Kareem writes the numbers 5 to 12 in the grid.

Show that for Kareem’s grid, the sum of the numbers in the top half of the grid is 16 less

than the sum of the numbers in the bottom half of the grid.

Use algebra to prove that for any set of eight consecutive numbers written in this grid in the same way, the sum of the numbers in the top half of the grid is 16 less than the sum of the

numbers in the bottom half of the grid.

Choose a word from this list which best describes each statement.

Equation Expression Formula Inequality Term

2 b + 2 w
(a)........

πr² = 30
(b)........

The area of this rectangle can be written as ax² + bx - 10.

Find the values of a, b and c. You must show your working.

Complete this identity by writing in the missing numbers.

4 (........ x + 1) = 14 x - 6 (x - 2) -........

Darcy says that the statement in the box is an equation. Ellis says that the statement in the box is an identity. One of them is correct.

Explain which one of Darcy or Ellis is correct.

........ is correct because........

Solve by factorising.

x² + 4x - 12 = 0

(x + 2)(3x + a)(bx + 3) = 6x³ + 11x² - 17x - 30

Find the value of a and the value of b.

Use algebra to prove that an odd number multiplied by a different odd number always gives an

answer that is an odd number.

Here are some algebraic statements.

v = u + at a + 2 b 3(x + 2) = 3 x + 6 2 y < x 2 x = 5

From the list above, write down an example of each of the following.

An expression.

An inequality.

An equation.

In this question, all measurements are in centimetres.

The square and the rectangle have the same area.

Show that x² - 8x - 20 = 0.

Solve x² - 8x - 20 = 0.

Explain why one of the answers in part (b) is not possible in the context of the question.

Write down the following.

The area of the square.

The length of the rectangle.

n is a positive integer.

Prove that (2 n + 1)(n - 3)(n + 2) + 3 n (n + 7) is always even.

The following kinematics formulas may be used in this question.

v = u + at

v² = u² + 2as

The initial velocity of a particle is 20 m/s. The acceleration of the particle is -8 m/s². After t seconds, the particle has travelled 25 m.

Show that 4t² - 20t + 25 = 0.

Solve 4t² - 20t + 25 = 0.

Show that the particle is stationary when it has travelled 25 m.

Choose a word from this list that best describes each statement.

Identity Expression Formula Term Equation

8 = n + 2
(a)........

3 x + 2 y
(b)........

(a + b)(a - b) = a² - b² (c)........

The diagram shows a square.

(4 x – 10) cm
Not to scale

(11 – 2 x) cm

By setting up and solving an equation, show that the perimeter of the square is numerically equal to the area of the square.

5 (2 x + 1) + c (x + d) = 12 x - 1

Work out the value of c and the value of d.

Here are two functions.

Jo chooses a number, x. She inputs x into each function. The two outputs are equal.

Work out the value of x.

Explain why there is no other input that gives two outputs that are equal.

Here is function C.

Kai chooses values for p and q so that if he inputs any number into both function A and function C, he will always get two outputs that are equal.

Find the value of p and the value of q.

Nine consecutive numbers are written on a 3-by-3 grid. They are arranged, in ascending order, in a spiral as shown.

Karen writes the numbers 3 to 11 on her grid.

The total of the first column is 3 + 10 + 9 = 22.

Karen says

The total of the first column is one less than the total of the second column.

Show that this is correct for Karen’s grid.

Victor says

If any nine consecutive numbers are arranged in ascending order in this spiral on a 3-by-3 grid, the total of the first column will always be one less than the total of the second column.

Prove that Victor is correct.

Prove that the mean of any four consecutive even integers is an integer.

Prove that 2x + 13x + 2 + x3x + 5 + 2 is a perfect square.

Gemma says The equation (2x + 1)(3x + 2) + x (3x + 5) + 2 = - 12 has no solutions. Explain Gemma’s reasoning.

This shape consists of three semicircles.

The length of PQ is 4 x cm.

Show that the area, in cm², of the whole shape is 3x².

n is a positive integer.

Prove that 13 n + 3 + (3 n - 5)(2 n + 3) is a multiple of 6.

(x + a)(x + 3)(2x + 1) = bx³ + cx² + dx - 12

Find the value of a, b, c and d.

n is an integer.

Explain why 2 n + 1 is an odd number.

Write down an algebraic expression for the next odd number after 2 n + 1.

Use algebra to show that the sum of two consecutive odd numbers will always be a

multiple of 4.

The middle number of three consecutive whole numbers is 2 a.

Prove that the sum of these three numbers cannot be 250.

Prove that the difference between two consecutive square numbers is always odd.

The diagram shows a cross placed on a number grid.

L is the product of the left and right numbers of the cross. T is the product of the top and bottom numbers of the cross. M is the middle number of the cross.

Show that when M = 35, L – T = 99.

Prove that, for any position of the cross on the number grid above, L – T = 99.

n is an integer.

Explain why 2 n + 1 is an odd number.

Prove that the difference between the squares of two consecutive odd numbers is a multiple of 8.