Math · Unit 12
Algebra
Linear equations, systems, polynomials and quadratics
Algebra proper is about manipulating relationships rather than evaluating them, and about recognising the small number of standard forms that most problems can be pushed into.
The unit covers linear equations, systems of equations, exponent laws, polynomials and factoring, and quadratics - including why the quadratic formula works rather than only how to apply it.
This unit breaks down into 21 short steps and 121 questions, starting at difficulty 3 and building to 4. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.
- Steps
- 21
- Questions
- 121
- Difficulty
- 3-4
What this unit covers
- Linear Equations
- Polynomials and Factoring
- Quadratic Equations
- Systems of Equations
- Exponent Laws
Where this fits
Needs Pre-Algebra. Required for Functions and Graphs and everything after it.
Where people slip
Factoring and expanding are inverse operations, and most algebra errors are one performed when the other was needed. Check which direction the problem wants before starting.
How the unit is structured
Algebra runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 3 and climb to 4 as you progress.
Challenge rounds
Example questions
30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 121 in the unit altogether.
Exponent Laws
- Build the sentenceLevel 3
1. Build a true statement about exponents.
Answer: Any number raised to the zero power equals one
Any nonzero number raised to the zero power equals one.
- Choose all that applyLevel 3
2. Select all expressions that equal x^8.
- x^5 x x^3correct
- (x^4)^2correct
- x^4 x x^4correct
- x^10 / x^5
x^5 x x^3, (x^4)^2, and x^4 x x^4 all give x^8, but x^10 / x^5 = x^5.
- Fact or fibLevel 3
3. x^5 / x^2 = x^3.
Answer: True
When dividing powers with the same base you subtract exponents: 5 - 2 = 3, so it is true.
- Fill the blankLevel 3
4. By the product rule, x^4 x x^3 = x^____.
- 7correct
- 12
- 1
- 43
Multiplying powers with the same base adds the exponents: 4 + 3 = 7.
- Match the pairsLevel 3
5. Match each expression to its simplified power of x.
Answer: x^5 x x^2 = x^7; x^8 / x^3 = x^5; (x^2)^5 = x^10
Add exponents when multiplying and subtract when dividing to simplify each one.
- Multiple choiceLevel 3
6. What is 2^3 x 2^2?
- 32correct
- 64
- 16
- 36
Add the exponents: 2^3 x 2^2 = 2^5 = 32.
Linear Equations
- Build the sentenceLevel 3
7. Build a true statement about linear equations.
Answer: A linear equation has one solution
A linear equation in one variable has exactly one solution.
- Choose all that applyLevel 3
8. Select all equations that are equivalent to 2x = 10.
- x = 5correct
- 4x = 20correct
- x + x = 10correct
- x = 10
x = 5, 4x = 20, and x + x = 10 all mean x = 5, but x = 10 does not.
- Fill the blankLevel 3
9. To solve 2x - 4 = 10, first add 4 to get 2x = ____, then divide by 2.
- 14correct
- 6
- 7
- 12
Adding 4 to 10 gives 2x = 14, so x = 7.
- Guess the numberLevel 3
10. Solve for x: 5x - 3 = 22. What is x?
Answer: 5
Add 3 to get 5x = 25, then divide by 5 to find x = 5.
- Fact or fibLevel 4
11. The equation 3(2x - 4) = 6x - 12 is true for every value of x.
Answer: True
Expanding the left side gives 6x - 12, which matches the right side, so it holds for all x.
- Match the pairsLevel 4
12. Match each equation to its solution.
Answer: x + 5 = 12 = x = 7; 2x = 6 = x = 3; 4x = 20 = x = 5
Isolate x in each equation to find its solution.
Polynomials and Factoring
- Fact or fibLevel 3
13. x^2 - 9 factors as (x - 3)(x + 3).
Answer: True
This is a difference of squares, and (x - 3)(x + 3) = x^2 - 9, so it is true.
- Fill the blankLevel 3
14. The expression x^2 + 8x + 15 factors as (x + 3)(x + ____).
- 5correct
- 12
- 8
- 2
You need factors of 15 that add to 8, which are 3 and 5, so the missing number is 5.
- Match the pairsLevel 3
15. Match each squared binomial to its expanded form.
Answer: (x + 1)(x + 1) = x^2 + 2x + 1; (x + 2)(x + 2) = x^2 + 4x + 4; (x + 3)(x + 3) = x^2 + 6x + 9
Squaring (x + n) gives x^2 + 2nx + n^2 for each pair.
- Multiple choiceLevel 3
16. Expand (x + 2)(x + 3).
- x^2 + 5x + 6correct
- x^2 + 6x + 6
- x^2 + 5x + 5
- x^2 + 6x + 5
Using FOIL: x^2 + 3x + 2x + 6 = x^2 + 5x + 6.
- Build the sentenceLevel 4
17. Build a true statement about factoring.
Answer: Factoring reverses the process of expanding
Factoring reverses the process of expanding a product of polynomials.
- Choose all that applyLevel 4
18. Select all expressions that are factored correctly.
- x^2 - 4 = (x - 2)(x + 2)correct
- x^2 + 6x + 9 = (x + 3)^2correct
- x^2 + 2x + 1 = (x + 1)^2correct
- x^2 - 1 = (x - 1)(x - 1)
x^2 - 4, x^2 + 6x + 9, and x^2 + 2x + 1 are factored right, but x^2 - 1 should be (x - 1)(x + 1).
Quadratic Equations
- Build the sentenceLevel 3
19. Build a true statement about quadratic equations.
Answer: A quadratic equation can have two solutions
A quadratic equation can have two solutions where its graph crosses the x-axis.
- Fill the blankLevel 3
20. The quadratic x^2 - 4 = 0 has solutions x = 2 and x = ____.
- -2correct
- 4
- 0
- -4
Since (-2)^2 = 4, the other solution is x = -2.
- Multiple choiceLevel 3
21. Solve x^2 - 5x + 6 = 0.
- x = 2 or x = 3correct
- x = -2 or x = -3
- x = 1 or x = 6
- x = 2 or x = -3
It factors as (x - 2)(x - 3) = 0, so x = 2 or x = 3.
- Choose all that applyLevel 4
22. Select all values of x that satisfy x^2 - x - 6 = 0.
- x = 3correct
- x = -2correct
- x = 2
- x = -3
It factors as (x - 3)(x + 2) = 0, so the solutions are x = 3 and x = -2.
- Fact or fibLevel 4
23. The equation x^2 + 4 = 0 has two real solutions.
Answer: False
This gives x^2 = -4, and no real number squares to a negative, so it has no real solutions.
- Match the pairsLevel 4
24. Match each quadratic equation to its solutions.
Answer: x^2 - 1 = 0 = x = 1 or x = -1; x^2 - 4 = 0 = x = 2 or x = -2; x^2 - 9 = 0 = x = 3 or x = -3
Each is a difference of squares, so the roots are plus and minus the square root of the constant.
Systems of Equations
- Build the sentenceLevel 3
25. Build a true statement about two intersecting lines.
Answer: Two lines that cross share one point
Two lines that cross share exactly one point, which is the solution of the system.
- Choose all that applyLevel 3
26. The system x + y = 6 and x - y = 2 has solution x = 4, y = 2. Select all true statements.
- x = 4correct
- y = 2correct
- The solution is (4, 2)
- x = 2
x = 4, y = 2, and the ordered pair (4, 2) all describe the solution; x = 2 is not correct.
- Fact or fibLevel 3
27. A system of two linear equations can have exactly one solution.
Answer: True
When two lines cross at a single point, the system has exactly one solution, so it is true.
- Fill the blankLevel 3
28. When solving a system by elimination, you add or ____ the equations to cancel a variable.
- subtractcorrect
- multiply only
- graph
- factor
Subtracting (or adding) the equations removes one variable so you can solve for the other.
- Match the pairsLevel 3
29. Match each system to its solution (x, y).
Answer: x + y = 5, x - y = 1 = (3, 2); x + y = 7, x - y = 1 = (4, 3); x + y = 9, x - y = 3 = (6, 3)
Solving each system by adding the equations gives the listed ordered pair.
- Multiple choiceLevel 3
30. For the system x + y = 10 and x - y = 4, what is x?
- x = 7correct
- x = 3
- x = 5
- x = 6
Adding the two equations gives 2x = 14, so x = 7 (and y = 3).
Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.
How you practise
This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.
- Build the sentence
- Choose all that apply
- Fact or fib
- Fill the blank
- Guess the number
- Match the pairs
- Multiple choice
- Odd one out
- Picture question
- Put in order
- Sort into groups
- Spell it
- True or false
- Type the answer
Practise Algebra
121 questions across 21 steps. Start with step one and crawl at your own pace.
Play this unitRead about Algebra
Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.
- How to Solve Linear Equations Step by StepLearn how to solve linear equations by keeping both sides balanced, undoing operations in order, and checking your answer with simple examples.August 13, 2026 · 5 min read
- How to Factor Quadratics Step by StepLearn how to factor quadratics by finding number pairs, using the product-sum method, checking your brackets, and spotting cases that need other methods.August 17, 2026 · 6 min read
- Quadratic Formula: How to Solve Quadratic EquationsThe quadratic formula gives you a reliable way to solve equations in standard form. Learn each part, avoid sign errors, and see what the discriminant tells you.August 14, 2026 · 6 min read
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