Completing the Square and Algebraic Fractions
Reveal quadratic structure by completing the square and operate on algebraic fractions with valid restrictions.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
A unitary quadratic x² + bx + c can be written as (x + p)² + q by halving the x coefficient and correcting the constant.
Exam cue: Expand the completed-square form to check p and q.
Concept 2
Algebraic fractions simplify by factorising and cancelling common factors, not individual terms joined by addition.
Exam cue: Factor every numerator and denominator before cancelling.
Concept 3
The four operations follow numerical fraction rules while denominators must remain non-zero.
Exam cue: For addition or subtraction, build a common denominator before combining numerators.
Risk pitfalls and guardrails
Forgetting the constant correction after making the square.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Cancelling terms across addition or subtraction.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Dividing fractions without inverting the divisor.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Complete the square
Half the x coefficient, square it, then correct the constant.
Square-form check
Expand (x + p)² + q to recover the original quadratic.
Algebraic cancellation
Cancel common factors, never separate added terms.
Fraction addition
Use a common denominator before combining numerators.
Fraction division
Multiply by the reciprocal.
Restriction
Any denominator must be non-zero.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Write x² + 6x + 2 in completed-square form.
Complete the square for x² − 8x + 1.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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