More about Equations
Master linear-quadratic simultaneous equations, unmask hidden quadratic equations in disguise, and learn how to systematically check your roots to avoid losing easy marks.
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The Core Concept
The Power of Substitution: Solving complex equations is all about unmasking the underlying structure. When solving a linear-quadratic system, we substitute the linear "clue" to reduce two variables into a single quadratic equation. When solving equations in disguise, we use the "Let u = …" substitution trick to transform fractional, exponential, logarithmic, or trigonometric terms into a standard quadratic form (
au² + bu + c = 0).DSE More about Equations Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Linear-Quadratic | Make y (or x) the subject of the linear equation and substitute it into the quadratic equation:ax² + bx + c = 0 | Intersection of a straight line and a curve (parabola or circle). Do not plug x back into the quadratic: Always substitute solved x values back into the linear equation to find y to avoid pairing errors. |
| 2. Fractional | Multiply the entire equation by the LCM of the denominators to clear the fractions. | Analyzing rational curves. Always check your denominators: Any solution that makes a denominator equal to 0 must be rejected. |
| 3. Exponential | Let u = aˣ to transform:p(a²ˣ) + q(aˣ) + r = 0 ⟹ pu² + qu + r = 0 | Analyzing rapid growth curves. Reject negative u values: Since aˣ > 0 for all real x, any negative solution (e.g., 2ˣ = −1) has no real solution. |
| 4. Logarithmic | Condense log terms using laws first:log M + log N = log(MN)If log M = log N ⟹ M = N | Analyzing logarithmic curves. Verify the domain strictly: The argument of any log must be strictly positive. Reject any roots that make an argument ≤ 0. |
| 5. Trigonometric | Let u = sin θ (or cos θ) to transform:pu² + qu + r = 0 | Analyzing repeating wave curves. Reject u outside [−1, 1]: Since −1 ≤ sin θ ≤ 1 and −1 ≤ cos θ ≤ 1, reject any u value outside this range, then solve for θ. |
💡 Exam Traps & Secrets
⚠️ The Linear-Quadratic Pairing TrapWhen you solve a linear-quadratic system, you will obtain two values for x (e.g.,
The Trap: If you substitute them into the quadratic equation, you will generate extra, incorrect coordinate pairs and lose the final accuracy marks.
x = 2 or x = −1). You must plug these values back into the linear equation to find the corresponding y values.The Trap: If you substitute them into the quadratic equation, you will generate extra, incorrect coordinate pairs and lose the final accuracy marks.
⚠️ The Logarithmic Domain CheckWhen you solve a logarithmic equation, you must check your final solutions against the original log arguments.
Example: If you solve
At
At
Example: If you solve
log(x − 2) + log(x + 1) = log 10 and obtain x = 4 or x = −3:At
x = 4 ⟹ log(4 − 2) = log(2) (valid).At
x = −3 ⟹ log(−3 − 2) = log(−5) (invalid!). You must reject the extraneous root x = −3.⚠️ The "Does Not Exist" / Tangency ConditionIf a question asks for the condition where a line and a curve touch at exactly one point (or are tangent), substitute to form a quadratic and set the discriminant to zero:
Δ = b² − 4ac = 0⚠️ Word Problem Boundary ChecksWhen translating real-world problems (e.g., rectangle dimensions) into quadratic equations, check if your answers make physical sense. Dimensions, lengths, times, or counts cannot be negative. Always reject negative solutions and write a clear concluding sentence.
✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Linear-Quadratic substitution & coordinate pairing)Solve the simultaneous equations:
y − x = 1 and y = x² − 1.(Quadratic substitution with u = 2ˣ, rejecting negative roots)Solve the exponential equation
4ˣ − 3 · 2ˣ − 4 = 0.(Condensing log terms, quadratic factorization, rejecting extraneous roots)Solve the logarithmic equation
log(x − 2) + log(x + 1) = log 10.(Δ = 0)Find the value(s) of k for which the line
y = kx + 3 and the parabola y = x² − 2 have exactly one solution.More about Equations Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
More about Equations · Form 4
128 MC questions · 35 long questions
Frequently Asked Questions
Use elimination for two linear equations (multiply and add/subtract to cancel a variable). Use substitution when one equation is already solved for a variable, or one is linear and the other quadratic. For linear-quadratic systems, substitution into the linear equation is mandatory to avoid the plug-back trap.
After substituting the linear equation into the quadratic, you get a quadratic in one variable. The discriminant Δ = b² − 4ac tells you how many real solutions exist: Δ > 0 means 2 intersections, Δ = 0 means the line is tangent (1 intersection), Δ < 0 means no intersection. This is frequently tested in Paper 2 and Section B.
Logarithms require strictly positive arguments (> 0); fractional denominators cannot be zero. After solving, substitute every solution back into the original expression. Reject any root that violates these conditions — this is called rejecting extraneous roots and failing to do so costs the final accuracy mark in Paper 1.