Quadratic simultaneous equations are systems where at least one equation contains non‑linear terms such as , , or . These systems generally model relationships where a curve intersects another curve or a line, leading to one or more solutions depending on how the graphs meet.
Simultaneous equations require finding values of and that satisfy both equations at the same time, meaning solutions represent intersection points of their graphs. This interpretation helps visualize why solutions may vary from zero to multiple depending on whether the graphs meet.
Linear–quadratic systems arise most commonly, involving one line and one curve. These are generally easier to solve because the linear equation can be rearranged efficiently, making substitution straightforward and avoiding unnecessary algebraic complexity.
Rearranging the linear equation into explicit form (such as or ) simplifies substitution because linear expressions isolate variables cleanly. This minimises algebraic errors and keeps later quadratic expansion manageable.
Substituting into the quadratic allows the curved relationship to incorporate the linear constraint, producing a single‑variable quadratic. Expanding and simplifying this expression creates a standard form that can be solved using factorisation or the quadratic formula.
Back‑substitution connects each root of the quadratic with its corresponding value in the linear equation. This ensures that each pair is accurately matched, avoiding mismatched answers that fail to satisfy the system.
| Feature | Linear Substitution | Quadratic Substitution |
|---|---|---|
| Resulting equation | Always linear | Becomes quadratic |
| Number of solutions | At most one | Up to two real solutions |
| Interpretation | Intersection of two lines | Intersection of line and curve |
Choosing the substitution direction matters because substituting a quadratic into a linear typically increases complexity unnecessarily. Using the linear equation as the substitution source keeps algebraic manipulation simple and reduces error risk.
Equation structure differences influence the difficulty of solving. Linear systems remain in first‑degree expressions, while quadratic systems require expansion and potentially the quadratic formula, making careful simplification essential.
Always rearrange the linear equation first, because this ensures that the substituted expression is as simple as possible. This reduces expansion steps and lowers the chance of missing terms during algebraic manipulation.
Check the number of solutions by inspecting the discriminant after forming the quadratic. This lets you anticipate whether to expect two, one, or no real intersection points, helping verify that your working is reasonable.
Write paired solutions clearly, ensuring each corresponds to its correct . Examiners often deduct marks for unpaired or mismatched answers even when individual values are correct.
Incorrect expansion of squared binomials such as often leads to missing the middle term. Recognising that prevents algebraic inaccuracies that can derail the entire solution.
Assuming every system has two solutions is a misconception because the quadratic may have a repeated root or no real root. Understanding how the discriminant shapes possible outcomes avoids false conclusions and signals when to recheck algebra.
Forgetting to test solutions against both equations can allow extraneous or incorrect roots to go unnoticed. Verifying each pair ensures that arithmetic or substitution mistakes are detected early.
