Algebra (LC Ordinary Level) — Key Notes & Examples
Concise notes aligned with recent LC OL papers (2019–2025), with one representative example per skill.
- Use the distributive law: multiply each term in the linear factor by each term in the quadratic, then collect like terms.
Example. Expand and simplify \((x+3)(2x^2 - x + 4)\).
Area model (2 × 3): rows correspond to \(x\) and \(+3\); columns correspond to \(2x^2,\ -x,\ +4\).
Combine like terms: \(2x^3 + (-x^2+6x^2) + (4x-3x) + 12 = 2x^3 + 5x^2 + x + 12\).
Answer: \(\boxed{2x^3 + 5x^2 + x + 12}\).
- Simplify carefully; collect like terms; isolate \(x\).
- For fractional equations, clear denominators using the LCM.
- Check domain restrictions if denominators depend on \(x\).
Example. Solve \( \dfrac{3x+1}{5}+\dfrac{x-2}{2}=\dfrac{47}{10} \) for \(x\in\mathbb{R}\).
- Standard form \(ax^2+bx+c=0\) \((a\neq0)\).
- Quadratic formula: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\).
- Factor where convenient; otherwise use the formula and round as requested.
Example. Solve \(3x^2-5x+1=0\) (to 2 d.p.).
- Maintain the inequality direction; if you multiply/divide by a negative, reverse it.
Example. Solve \(2x+4\ge 6x-8\) for \(x\in\mathbb{R}\).
- Use substitution or elimination; present a clear chain of working.
Example. Solve \(\begin{cases}3x+2y=11\\ x-4y=-1\end{cases}\).
- Substitute the linear relation into \(x^2+y^2=r^2\); solve the resulting quadratic.
Example. Solve \(\begin{cases}2x+y=5\\ x^2+y^2=25\end{cases}\).
- Find the quotient so that divisor × quotient = dividend (plus remainder, if any). The 2×2 area model shows the products that reconstruct the dividend.
Example. Simplify \( \dfrac{6x^2 - 23x + 20}{\,2x - 5\,} \).
Idea: Seek a quotient \(ax+b\) so that \((2x-5)(ax+b)=6x^2-23x+20\).
Check the middle term: \(-8x + (-15x) = -23x\). Hence \((2x-5)(3x-4)=6x^2-23x+20\).
\(\displaystyle \frac{6x^2 - 23x + 20}{2x - 5} = \boxed{3x - 4},\quad x \ne \tfrac{5}{2}.\)
- Treat letters as numbers; isolate the required variable step by step.
- State domain restrictions if division by a variable expression occurs.
Example. Write \(q\) in terms of \(p,t\) given \(3(2p+q)=t\).
- Substitute accurately; simplify to a single rational number where asked.
- Expand and collect like terms to rewrite \(f(x)\) in \(ax^2+bx+c\).
Example. Evaluate \( \dfrac{3x+5}{10}-\dfrac{1}{x+3}\) at \(x=2\) and express as \(\tfrac{a}{b}\) in lowest terms.
- Exponent laws: \(a^m a^n=a^{m+n}\), \((a^m)^n=a^{mn}\), \(a^{-n}=1/a^n\).
- Convert surds to fractional indices and simplify.
Example. Find \(x\in\mathbb{Q}\) such that \(2^x=\sqrt{32}\).
