Chapter Overview
🧭Your visual roadmap through the chapter
Expressions & Equations
Terms, variables, constants, coefficients, degree of a polynomial
Solving Linear Equations
One-variable & simultaneous two-variable equations; word problems
Quadratic Equations
Standard form, graph shape, roots as x- intercepts
Solving Quadratics
Factorisation, completing the square, the quadratic formula
Building & solving algebraic equations of degree 1 (linear) and degree 2 (quadratic), including systems of two linear equations, using factorisation, completing the square and the quadratic formula.
Equation-solving underpins later chapters — coordinate geometry, linear programming, financial mathematics and break-even/optimisation problems all build directly on this chapter.
Translate word problems into equations; solve simultaneous equations by elimination; factorise, complete the square, and apply the quadratic formula accurately.
Key Definitions
📘Source-based terminology, exam-ready
Expression
A number, a variable, or a combination of numbers and variables with operation symbol(s); separated into terms by operations.
Key word: terms
Equation
Formed when two mathematical expressions are considered equal, shown by the sign "=".
Key word: equality
Variable / Constant
Variable: a value that may change within the scope of the problem (x, y, z). Constant: a value that remains unchanged, even if unknown (a, b, c).
Key word: changes vs fixed
Coefficient
A multiple attached to a variable within an equation (e.g. the 2 in y = 2x + c).
Key word: multiplier
Degree (of a term/equation)
In a polynomial, the sum of all exponents of the variables used; the degree of an equation is its highest power.
Key word: highest power
Linear Equation
Each term is a constant or the product of a constant and a single variable; no term is raised to a power or root. Degree = 1.
Key word: degree one
Dependent vs Independent Variable
y is the dependent variable (changes because of x); x is the independent variable. By convention, the dependent variable sits alone on the left of "=".
Key word: y depends on x
Quadratic Expression
An expression in which the degree of the variable is two (the highest power is a square).
Key word: degree two
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Linear Equations
Degree-one equation; standard straight-line form y = a + mx.
Model relationships that change at a constant rate.
Single unknown problems and simultaneous two-variable systems solved by elimination.
CONCEPT → Quadratic Equations
Degree-two equation, standard form y = ax2 + bx + c.
Find the two roots (x-values) where the curve crosses y = 0.
Area/distance, forecasting, interest, price-product modelling, optimisation (min/max).
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Solving a linear equation
Read the question; note requirements & given information
List the unknown variable(s), assign a letter (x)
Express the relationship between variables as an equation
Solve the equation for the unknown
Recheck the solution
RULE 3 Simultaneous linear equations
Manipulate one equation so a variable's coefficient matches the other equation
Add or subtract the equations to eliminate that variable
Solve for the remaining variable
Substitute back into an original equation to find the other variable
RULE 2 Formulating from word problems
Step 1: Pick the smaller/unknown value as x
Step 2: Write other information in terms of x
Step 3: Cancel out written values; keep remaining info
Step 4: Formulate the equation and solve
Step 5: Check the solution
RULE 4 Discriminant (b²−4ac) conditions
If b² − 4ac > 0 → roots are real and unequal
If b² − 4ac < 0 → roots are imaginary / complex
If b² − 4ac = 0 → roots are real and equal
Not every quadratic can be factorised — use completing the square or the formula instead
Formula Zone
🧮Every formula supported by the source text
STRAIGHT LINE (LINEAR EQUATION)
y = dependent variable x = independent variable a = y-intercept m = slope/gradient
QUADRATIC EQUATION — STANDARD FORM
a, b, c are constants; a ≠ 0. Plots as a U-shape (a > 0) or inverted-U (a < 0).
QUADRATIC FORMULA
Use when a x2 + b x + c = 0. Constant must be on the RHS → general form first. Automatically "completes the square" for you.
PERFECT-SQUARE IDENTITIES (FOR COMPLETING THE SQUARE)
Reference identities used to rewrite ax²+bx as a perfect square, then solve for x.
Calculation Roadmap
🗺️General path for any equation question
1· Read Question
Identify given information
2· Identify Unknown
Assign a letter (x)
3· Select Method
Formula, factorisation, elimination
4· Substitute
Put values into equation/formula
5· Calculate & Check
Solve, recheck, state final answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Simultaneous Linear Equations (Elimination)
- 1Multiply one/both equations so a variable's coefficient matches
- 2Add or subtract the equations to eliminate that variable
- 3Solve the single remaining equation for the other variable
- 4Substitute back into an original equation to find the first variable
METHOD C · Factorisation (coefficient of x² > 1)
- 1Multiply coefficient of x² by the constant (a × c)
- 2Find two numbers summing to b, multiplying to (a×c)
- 3Rewrite the x-term as two new x-terms using those numbers
- 4Bracket first two & last two terms (flip sign inside bracket if needed)
- 5Factor each bracket separately to find a common factor
- 6Complete the factorisation; set each bracket = 0 and solve
METHOD B · Factorisation (coefficient of x² = 1)
- 1Write in standard form x² + bx + c = 0
- 2Find two numbers that sum to b and multiply to c
- 3Write as (x ± ?)(x ± ?) = 0
- 4Set each bracket = 0 and solve for x
METHOD D · Completing the Square
- 1Rearrange ax²+bx+c=0 into ax²+bx = −c
- 2Divide throughout by a (coefficient of x² must be 1)
- 3Take half the coefficient of x (with its sign)
- 4Square that number
- 5Add it to both sides of the equation
- 6Rearrange the LHS into a perfect square and solve for x
METHOD E · The Quadratic Formula
- 1Write the equation in general form ax²+bx+c=0
- 2Identify a, b, and c (with correct signs)
- 3Substitute into x = (−b ± √(b²−4ac)) ÷ 2a
- 4Evaluate the discriminant first, then both root values
Different Cases / Scenarios
🔀Recognise the situation, pick the right route
IF the equation has one variable, degree 1
→ USE the 5-step linear solving approach (isolate x)
IF two linear equations share two unknowns
→ USE elimination (match coefficients, add/subtract)
IF the quadratic factorises easily (small integers)
→ USE factorisation — fastest method
IF the quadratic does not factorise neatly
→ USE completing the square or the quadratic formula
IF discriminant b²−4ac > 0
→ Roots are real and unequal (curve cuts x-axis twice)
IF discriminant b²−4ac = 0
→ Roots are real and equal (curve touches x-axis once)
IF discriminant b²−4ac < 0
→ Roots are imaginary/complex (curve never touches x-axis)
IF a word problem gives two unknowns, one described relative to the other
→ Let the smaller value = x, express the other in terms of x
Comparison Tables
⚖️Don't confuse these!
| Linear Equation | Quadratic Equation | |
|---|---|---|
| Degree | One (highest power = 1) | Two (highest power = 2) |
| Standard form | y = a + mx | y = ax² + bx + c |
| Graph shape | Straight line | U-shape or inverted-U (parabola) |
| Number of roots | One root (single-variable case) | Up to two roots (x-axis crossings) |
| Factorisation | Completing the Square | Quadratic Formula | |
|---|---|---|---|
| When to use | When factors are easy to spot | When it cannot be factorised neatly | Always works; used when other methods are difficult |
| Core idea | Reverse of expanding brackets | Rewrite as a perfect square using (a±b)² identities | Formula that auto-completes the square |
| Limitation | Not every quadratic factorises | Only works cleanly once coefficient of x² = 1 | None — but requires correct a, b, c signs |
Graph Interpretation
📈Reading the quadratic curve
Shape rule
If a > 0 → graph is U-shaped. If a < 0 → graph is an inverted U. Example: x²−8x+12=y cuts the x-axis at x = 2 and x = 6 — these are the equation's two roots.
Reading the roots
Roots = the x-values where the curve crosses the line y = 0 (the x-axis). Not every quadratic crosses the x- axis — if it doesn't, the equation is insoluble in real numbers.
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
Profit function: P(x) = −0.025x² + 9.85x − 60. Find profit at x = 0 (no units sold) and the break-even units.
GIVEN: a = −0.025, b = 9.85, c = −60 | At x=0: P(0) = −60 → loss of Rs. 60.
FORMULA: x = (−b ± √(b²−4ac)) ÷ 2a for break-even (P(x)=0)
SUBSTITUTION: x = (−9.85 ± √(9.85² − 4(−0.025)(−60))) ÷ 2(−0.025)
CALCULATION: x = (−9.85 ± √92.0225) ÷ −0.05 = (−9.85 ± 9.5406) ÷ −0.05
FINAL ANSWER: x ≈ −6.19 or x ≈ 387.81 → break-even at approximately 388 units
Exam Question Patterns
📝Definition/conceptual — identify degree, variable type, dependent/independent variable
Formula-based — apply the quadratic formula directly
Calculation — solve linear/quadratic equations by any method
Application/scenario — ages, geometry, cost/price, break-even word problems
Simultaneous-equation MCQs asking for a combined value (e.g. xy, 5x+2y)
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Sign errors in b²−4ac when a or c is negative — a double-negative becomes addition.
- ✗ Forgetting to flip the sign inside brackets when bracketing a subtracted term.
- ✗ Dividing by a in completing the square before rearranging — must rearrange to ax²+bx=−c first.
- ✗ Choosing the wrong unknown in word problems — always let x be the smaller/simplest value.
- ✗ Mismatched coefficients in elimination — multiply the whole equation, not just one term.
Quick Decision Tree
🌳Chapter 1 in 60 Seconds
⚡Linear: y = a + mx, degree 1
Quadratic: y = ax²+bx+c, degree 2
Simultaneous: match coefficients, eliminate
Formula: x=(−b±√(b²−4ac))÷2a
b²−4ac>0: real, unequal roots
b²−4ac=0: real, equal roots
b²−4ac<0: imaginary roots
Trap: always flip sign inside brackets
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