Algebra: Expressions and Equations

Algebra: Expressions and Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

An algebraic expression combines numbers, variables, and operations without an equals sign (e.g., 3x² + 5x − 7); the moment you write an = sign you have an equation. A linear equation ax + b = 0 solves to x = −b/a. A quadratic equation ax² + bx + c = 0 (a ≠ 0) solves via the quadratic formula x = [−b ± √(b² − 4ac)] / 2a, where the discriminant D = b² − 4ac decides root count. Key facts to memorise: sum of roots α + β = −b/a, product αβ = c/a, and the identity (a + b)² = a² + 2ab + b² — NOT a² + b². Transposition across = flips the sign. For HAT-UG, expect 3–5 direct MCQs: solving a linear equation, finding roots of a quadratic, or applying an exponent rule.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Expressions vs. Equations

An expression such as 5x − 3(x + 2) evaluates to a value once x is known but contains no =. An equation asserts two expressions are equal and is solved by finding the variable values that satisfy it. A polynomial of degree n has the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ with aₙ ≠ 0; degree 1 is linear, degree 2 is quadratic.

Solving Linear Equations

Isolate the variable by adding, subtracting, multiplying, or dividing both sides. Example: 4(x − 1) = 2x + 6 → 4x − 4 = 2x + 6 → 2x = 10 → x = 5. Every term crossing = changes sign (transposition).

Quadratic Equations

For ax² + bx + c = 0, the quadratic formula gives the roots. The discriminant D = b² − 4ac tells you:

• D > 0 → two distinct real roots
• D = 0 → one repeated real root (x = −b/2a)
• D < 0 → no real roots

Vieta’s formulas connect roots to coefficients: α + β = −b/a, αβ = c/a, useful for forming a quadratic when its roots are known.

Factorisation Techniques

Common techniques: pulling out a common factor, grouping terms, difference of squares a² − b² = (a + b)(a − b), and trinomial factoring x² + (p+q)x + pq = (x + p)(x + q). Factorising before applying the quadratic formula often reduces arithmetic.

Laws of Exponents

aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (for a ≠ 0), and a⁻ⁿ = 1/aⁿ. Note (a + b)² ≠ a² + b²; the expansion always includes the 2ab cross-term.

Common Question Types in HAT-UG

• Find the value of x in a linear equation.
• Solve a quadratic and state the nature of roots from D.
• Simplify an expression using exponent laws or factorisation.
• Set up an equation from a word problem (ages, profit, work-rate).

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Simultaneous Linear Equations in Two Variables

A pair like 2x + 3y = 12 and x − y = 1 is solved by substitution (express one variable, plug into the other) or elimination (multiply to cancel one variable). A unique solution exists when the coefficient ratios differ; parallel lines give no solution, coincident lines give infinitely many.

Inequalities

Replace = with <, >, ≤, or ≥. Solve exactly like linear equations, but multiplying or dividing by a negative number reverses the inequality sign — a frequent trap. Example: −3x < 9 → x > −3.

Edge Cases and Common Mistakes

• Dividing both sides by an expression that could be zero — you may discard the valid solution x = 0. Always factor first when possible.
• Forgetting to reverse the sign on −2(x − 4) > 10 type problems.
• Treating an expression like 3x + 5 as if it equals zero without justification.
• Misapplying (a + b)² as a² + b², a classic conceptual error tested on HAT-UG.
• Sign slip inside the quadratic formula: using +b instead of −b, or mis-handling −b ± √D.

Connections to Other Topics

Algebra feeds directly into functions and graphs (the quadratic’s parabola opens up if a > 0), sequences (roots of x² − x − 1 = 0 generate the golden ratio), and word problems in profit–loss, time–distance, and mixture questions that populate the Quantitative Reasoning section.

Worked Example

Solve 2x² − 7x + 3 = 0. Discriminant: D = 49 − 24 = 25. Roots: x = [7 ± 5] / 4, giving x = 3 or x = 1/2. Verification: α + β = 7/2 = −b/a ✓, αβ = 3/2 = c/a ✓.

Practice Prompts

1. If x² − 5x + k = 0 has equal roots, find k using the discriminant condition D = 0.
2. Simplify (3x²y⁻¹)² / (9xy⁻³) and state the value of the exponent of y.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.