Course

Cambridge O Level Mathematics: Algebra (4024)

Name: Cambridge O Level Mathematics: Algebra (4024)
Price: 1699 BDT

The whole 4024 algebra section from the first letter: expressions and substitution, expanding and factorising, algebraic fractions, indices, every kind of equation, inequalities, sequences and proportion.

100 lessons17h 23mEnglish

What you'll learn

Use letters for numbers, write expressions correctly and substitute any number into a formula
Expand up to three brackets, factorise every form in the syllabus and complete the square
Simplify, multiply, divide, add and subtract algebraic fractions
Use positive, zero, negative and fractional indices, and solve equations with indices
Construct and solve linear, fractional, simultaneous and quadratic equations, with answers in surd form when asked
Change the subject of a formula, including when the subject appears twice or under a power or root
Solve linear inequalities, show them on a number line and describe a region on a graph with inequalities
Find and use the nth term of linear, quadratic, cubic and exponential sequences
Write direct and inverse proportion as an equation and use it to find unknown values

Requirements

Number at O Level standard, or our Number course
Comfortable with negative numbers, fractions and the order of operations

Course content

100 lessons · 17h 23m

2.1 · Introduction to algebra

Letters stand for numbers you do not know yet. Write expressions the way examiners do, then put numbers in, including negatives and fractions, and use a formula to get an answer.

0/8

Using letters to represent numbers

A letter stands for any number. n packets of crisps at ৳20 each cost 20n taka, whatever n is.

Preview10m 2.

Writing algebraic expressions

Terms, expressions and the short way to write them: 3n, n², ab and a over b.

Preview10m

Substituting numbers into expressions

Put the number in place of the letter, then follow the order of operations.

10m

Substituting negative numbers and fractions

Use brackets round a negative number, so (-3)² is 9 and not -9.

11m

Substituting into formulas

A formula links quantities. Put in the ones you know to find the one you want.

11m

Introduction to algebra: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Introduction to algebra: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

Introduction to algebra: summary

The key points, rules and common mistakes of the chapter on one screen.

2.2 · Expanding and simplifying

Collect like terms, multiply and divide terms, then expand one, two and three brackets. A rectangle cut into pieces shows where every term comes from.

0/8

Simplifying expressions by collecting like terms

Only like terms can be added: 3a + 5a is 8a, but 3a + 5b stays as it is.

Preview10m 2.

Multiplying and dividing algebraic terms

Multiply the numbers, then the letters: 3a × 4ab = 12a²b.

Preview10m

Expanding a single bracket

The term outside multiplies every term inside: 3(x + 2) = 3x + 6.

11m

Expanding two brackets

Every term in the first bracket multiplies every term in the second, so (x + 3)(x + 2) has four pieces.

12m

Expanding three brackets

Expand two brackets first, then multiply that answer by the third bracket.

12m

Expanding and simplifying: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Expanding and simplifying: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

Expanding and simplifying: summary

The key points, rules and common mistakes of the chapter on one screen.

2.2 · Factorising and completing the square

Factorising is expanding in reverse. Take out common factors, group in pairs, spot the difference of two squares and perfect squares, factorise quadratics and cubics, then complete the square.

0/12

Factorising by extracting common factors

Take out the highest common factor, so 9x² + 15xy = 3x(3x + 5y). Factorise fully.

Preview11m 2.

Factorising by grouping

Four terms in two pairs: ax + bx + kay + kby = (a + b)(x + ky).

Preview11m

Factorising the difference of two squares

a² - b² = (a - b)(a + b), and it turns 99² - 1 into easy arithmetic.

11m

Factorising perfect squares

a² + 2ab + b² = (a + b)². Check that the middle term is twice the product.

10m

Factorising quadratic expressions: x² + bx + c

Find two numbers that multiply to c and add to b.

12m

Factorising quadratic expressions: ax² + bx + c

Find two numbers that multiply to ac and add to b, split the middle term, then group.

12m

Factorising expressions of the form ax³ + bx² + cx

Take out x first, then factorise the quadratic that is left.

11m

Completing the square

Halve the coefficient of x: x² + 6x + 5 = (x + 3)² - 4.

12m

Completing the square when a is not 1

Take a out of the x terms first: 2x² + 12x + 7 = 2(x + 3)² - 11.

12m

10.

Factorising and completing the square: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

11.

Factorising and completing the square: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

12.

Factorising and completing the square: summary

The key points, rules and common mistakes of the chapter on one screen.

2.3 · Algebraic fractions

Algebraic fractions follow the same rules as number fractions. Cancel common factors, never terms, then multiply, divide, add and subtract, and write the answer as a single fraction.

0/9

Simplifying algebraic fractions

Cancel a common factor from the top and the bottom. You can never cancel a term that is added.

Preview11m 2.

Factorising and simplifying rational expressions

Factorise the top and the bottom first. Only then can a whole bracket cancel.

Preview12m

Multiplying algebraic fractions

Multiply the tops and multiply the bottoms, cancelling common factors first.

10m

Dividing algebraic fractions

Turn the second fraction upside down and multiply.

10m

Adding and subtracting algebraic fractions: number denominators

Make the denominators the same, then add the tops: x/3 + x/2 = 5x/6.

12m

Adding and subtracting algebraic fractions: algebraic denominators

Multiply the denominators to get a common one, and put brackets round every numerator.

13m

Algebraic fractions: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Algebraic fractions: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

Algebraic fractions: summary

The key points, rules and common mistakes of the chapter on one screen.

2.4 · Indices II

The rules of indices with letters, what a zero, negative or fractional index means, simplifying expressions like (2x⁵/3)³, and solving equations such as 3²ˣ = 27 by matching the base.

0/8

The rules of indices

Multiply: add the indices. Divide: subtract them. Power of a power: multiply them.

Preview11m 2.

Zero and negative indices

Any number to the power 0 is 1, and x⁻² means 1 over x².

Preview11m

Fractional indices

The bottom of the fraction is a root and the top is a power: 8^(2/3) = 4.

12m

Using the rules of indices to simplify

Deal with the numbers and each letter separately, one rule at a time.

12m

Solving equations with indices

Write both sides as powers of the same number, then make the indices equal.

12m

Indices II: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Indices II: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

Indices II: summary

The key points, rules and common mistakes of the chapter on one screen.

2.5 · Linear equations

Write expressions, equations and formulas from words, then solve linear equations: one step, unknowns on both sides, brackets, and fractions with numbers or letters underneath.

0/10

Constructing expressions and formulas

Turn words into algebra: the product of two consecutive even numbers is 2n(2n + 2).

Preview11m 2.

Solving linear equations

Do the same thing to both sides until the letter is on its own: 3x + 4 = 10 gives x = 2.

Preview11m

Linear equations with the unknown on both sides

Collect the letter terms on one side and the numbers on the other.

11m

Linear equations with brackets

Expand the brackets first, then solve: 5 - 2x = 3(x + 7) gives x = -3.2.

11m

Constructing and solving linear equations

Choose a letter for the unknown, write the equation from the words, solve, then answer the question.

12m

Fractional equations with numerical denominators

Multiply every term by the lowest common denominator, then solve as usual.

12m

Fractional equations with algebraic denominators

Multiply both sides by the bracket underneath: 2x/(x + 1) = 4 gives x = -2.

12m

Linear equations: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Linear equations: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

10.

Linear equations: summary

The key points, rules and common mistakes of the chapter on one screen.

2.5 · Simultaneous linear equations

Two unknowns need two equations. Solve a pair by elimination, by elimination after multiplying, and by substitution, then build the pair yourself from a word problem.

0/8

Simultaneous linear equations

One receipt cannot fix two prices. A second receipt can, and the answer fits both equations.

Preview11m 2.

Solving simultaneous equations by elimination

When one pair of coefficients is the same, add or subtract the equations to remove that letter.

Preview12m

Elimination after multiplying an equation

Multiply one or both equations until a pair of coefficients match, then eliminate.

12m

Solving simultaneous equations by substitution

Make one letter the subject, then put that expression into the other equation.

12m

Constructing simultaneous equations

Give each unknown a letter, write one equation for each fact, then solve the pair.

12m

Simultaneous linear equations: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Simultaneous linear equations: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

Simultaneous linear equations: summary

The key points, rules and common mistakes of the chapter on one screen.

2.5 · Quadratic equations

Solve quadratic equations three ways: by factorisation, by completing the square and by the quadratic formula, with answers in surd form or to 2 decimal places, then build them from word problems.

0/10

Solving quadratic equations by factorisation

If two brackets multiply to 0, one of them is 0. That gives the two solutions.

Preview12m 2.

Solving quadratic equations: rearranging to equal zero first

Move every term to one side before you factorise. Never divide by x, or you lose x = 0.

Preview11m

Solving quadratic equations by completing the square

Write it as (x + p)² = q, then take the square root of both sides, plus or minus.

12m

Solving quadratic equations with the quadratic formula

Put a, b and c into the formula from the List of formulas. Give answers to 2 decimal places when asked.

13m

Quadratic equations: solutions in surd form

Exact answers keep the square root: x = 2 ± √3.

11m

Constructing and solving quadratic equations

Write the equation from the words, solve it, then reject a solution that makes no sense, such as a negative length.

13m

Fractional equations that lead to quadratic equations

Clear the denominators, collect terms to one side, and a quadratic appears.

13m

Quadratic equations: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Quadratic equations: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

10.

Quadratic equations: summary

The key points, rules and common mistakes of the chapter on one screen.

2.5 · Changing the subject of formulas

Rearrange a formula so a different letter is on its own: undo the operations in reverse order, clear fractions and brackets, deal with powers and roots, and factorise when the subject appears twice.

0/7

Changing the subject of a formula

Undo the operations in reverse order, doing the same to both sides, as when solving an equation.

Preview11m 2.

Changing the subject: fractions and brackets

Clear the fraction by multiplying both sides, or expand the bracket, before you rearrange.

Preview12m

Changing the subject: powers and roots

Square to undo a square root, and take a square root, plus or minus, to undo a square.

12m

Changing the subject: when the subject appears twice

Collect every term with the subject on one side, then factorise it out.

13m

Changing the subject of formulas: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

Changing the subject of formulas: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

Changing the subject of formulas: summary

The key points, rules and common mistakes of the chapter on one screen.

10.

2.6 · Inequalities

Show inequalities on a number line, solve linear inequalities, including the rule for multiplying by a negative number, then draw regions on a graph and list the inequalities that define a region.

0/10

Representing inequalities on a number line

An open circle for < and >, a closed circle for ≤ and ≥, and a line along every value that works.

Preview10m 2.

Solving linear inequalities

Solve it like an equation, and keep the inequality sign in every line: 3x < 2x + 4 gives x < 4.

Preview11m

Multiplying or dividing an inequality by a negative number

Multiplying or dividing both sides by a negative number reverses the inequality sign.

11m

Solving double inequalities

Do the same to all three parts: -3 ≤ 3x - 2 < 7 gives -1/3 ≤ x < 3. Then list the integers.

12m

Constructing and interpreting linear inequalities

Words like "at most" and "more than" become ≤ and >, and the answer is often a whole number.

12m

Linear inequalities in two variables

Draw the boundary line, broken for < or > and solid for ≤ or ≥, then shade the side you do not want.

13m

Representing a region with several inequalities

Draw each boundary line and shade each unwanted side. The region left clear satisfies them all.

13m

Listing inequalities that define a region

Find the equation of each boundary line, then test a point inside to choose the sign.

12m

Inequalities: practice

A timed drill of mixed questions from the whole chapter, each with a worked solution.

12m

10.

Inequalities: exam practice

Past-paper style questions with marks, full worked solutions and the mark scheme.

15m

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.