Algebra
Evaluating expressions, transposing formulae, indices, simultaneous and quadratic equations, number systems.
Algebra — summary notes
- Transposing (rearranging) a formula isolates the quantity you need. Whatever operation you apply to one side, apply to the other — that is the whole rule.
- Laws of indices let you combine powers without expanding them. A negative index means a reciprocal; a fractional index means a root.
- Simultaneous equations solve for two unknowns from two equations, by substitution or by elimination.
- A quadratic equation contains a squared term and generally has two solutions; the quadratic formula solves any of them.
- Logarithms convert multiplication into addition and are the basis of decibel and other logarithmic scales used in avionics.
- Binary and hexadecimal are the number systems behind digital aircraft systems — see Module 5 for their full treatment.
- Laws of indices
- aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
- Negative & fractional index
- a⁻ⁿ = 1 / aⁿ, a^(1/2) = √a, a⁰ = 1
- Quadratic formula
- x = [−b ± √(b² − 4ac)] / 2a
- Log laws
- log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a
Transpose V = IR to make R the subject.
Divide both sides by I: R = V / I. This is the form used whenever a resistance is calculated from a measured voltage and current.
Simplify (2x³)(3x⁴).
Multiply the coefficients and add the indices: 2 × 3 = 6 and x³⁺⁴ = x⁷, giving 6x⁷.
Solve x² − 5x + 6 = 0.
Factorising: (x − 2)(x − 3) = 0, so x = 2 or x = 3. Check with the formula: b² − 4ac = 25 − 24 = 1, x = (5 ± 1)/2 = 3 or 2. ✓
Algebra concept map
Algebra
Algebra quiz
Algebra
1. Simplify 3a + 2b − a + 4b.
2. Simplify 2(x + 3) + 3(x − 1).
3. Simplify 4x² + 3x².
4. Factorise 6x + 9.
5. Factorise x² + 5x.
6. Simplify x³ × x⁴.
7. Simplify x⁵ ÷ x².
8. Simplify (x²)³.
9. What is x⁰ (for non-zero x)?
10. Write x⁻¹ without a negative index.
11. Evaluate 2³ × 2².
12. The index form x^(1/2) means:
13. Simplify (2x²)³.
14. Solve 2x + 5 = 17.
15. Solve 3x − 4 = 11.