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Completing the Square
For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born.
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Completing the Square
What is it?
For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born.
In simple words
- For x^2 + px, add (p/2)^2 to make (x + p/2)^2.
- THE NUMBER TO ADD: x2 + 6x needs 32 = 9.
- SOLVE: x^2 - 4x + 1 = 0 -> (x-2)^2 - 4 + 1 = 0 -> (x-2)^2 = 3 -> x-2 = +/- sqrt(3) -> x = 2 +/- sqrt(3).
- Worked-method reminder: x = 2 + sqrt(3) or x = 2 - sqrt(3)
Important terms
COMPLETING THE SQUARE: For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born.
THE NUMBER TO ADD: x^2 + 6x needs 3^2 = 9. x^2 - 4x needs (-2)^2 = 4. Always square half of the x-coefficient (when a=1).
SOLVE: x^2 - 4x + 1 = 0 -> (x-2)^2 - 4 + 1 = 0 -> (x-2)^2 = 3 -> x-2 = +/- sqrt(3) -> x = 2 +/- sqrt(3).
What you should learn
- Explain completing the square using the correct definitions, conditions, and notation.
- Apply add (half of b)^2, then square root correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x^2+6x -> +9
- x^2-4x -> +4
- Not +6 or +16
THE NUMBER TO ADD
x^2 + 6x needs 3^2 = 9. x^2 - 4x needs (-2)^2 = 4. Always square half of the x-coefficient (when a=1).
SOLVE
x2 - 4x + 1 = 0 -> (x-2)2 - 4 + 1 = 0 -> (x-2)2 = 3 -> x-2 = +/- sqrt(3) -> x = 2 +/- sqrt(3).
Recommended for Class 10
Full school-level explanation
For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born.
Half, square, move:
WHAT — COMPLETING THE SQUARE
For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Completing The Square matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE NUMBER TO ADD, then SOLVE. Worked: solve x^2 - 4x + 1 = 0 by completing the square. x^2 - 4x + 1 = 0 x^2 - 4x = -1 half of -4 is -2, and (-2)^2 = 4 x^2 - 4x + 4 = -1 + 4 (x - 2)^2 = 3 x - 2 = +/- sqrt(3) x = 2 + sqrt(3) or x = 2 - sqrt(3) Finish by checking signs, restrictions, units, and whether the answer matches the question.
THE NUMBER TO ADD
x^2 + 6x needs 3^2 = 9. x^2 - 4x needs (-2)^2 = 4. Always square half of the x-coefficient (when a=1).
Remember
- (b/2)2 if a=1
Examples
- x^2+6x -> +9
- x^2-4x -> +4
- Not +6 or +16
SOLVE
x2 - 4x + 1 = 0 -> (x-2)2 - 4 + 1 = 0 -> (x-2)2 = 3 -> x-2 = +/- sqrt(3) -> x = 2 +/- sqrt(3).
Remember
- isolate the square
Examples
- (x-2)2 = 3
- x = 2 +/- sqrt(3)
- Two real roots
Remember
- add (half of b)^2
- then square root
- x = 2 +/- sqrt(3)
- make a = 1 first
- (b/2)2 if a=1
- isolate the square
- add (half of b)^2
- then square root
How the ideas connect
Worked: solve x^2 - 4x + 1 = 0 by completing the square. x^2 - 4x + 1 = 0 x^2 - 4x = -1 half of -4 is -2, and (-2)^2 = 4 x^2 - 4x + 4 = -1 + 4 (x - 2)^2 = 3 x - 2 = +/- sqrt(3) x = 2 + sqrt(3) or x = 2 - sqrt(3)
Deeper understanding
Go beyond memorising Completing the Square. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve x^2 - 4x + 1 = 0 by completing the square. x^2 - 4x + 1 = 0 x^2 - 4x = -1 half of -4 is -2, and (-2)^2 = 4 x^2 - 4x + 4 = -1 + 4 (x - 2)^2 = 3 x - 2 = +/- sqrt(3) x = 2 + sqrt(3) or x = 2 - sqrt(3)
Common mistakes
Exam focus
Core Idea
THE NUMBER TO ADD
SOLVE
Worked Method
Exam focus
Quick revision
- add (half of b)^2
- then square root
- x = 2 +/- sqrt(3)
- make a = 1 first
- COMPLETING THE SQUARE: For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadrati
- THE NUMBER TO ADD: x^2 + 6x needs 3^2 = 9. x^2 - 4x needs (-2)^2 = 4. Always square half of the x-coefficient (when a=1
If a is not 1, divide the whole equation by a first, then complete the square.
For long answers on Completing the Square: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- add (half of b)^2
- then square root
- x = 2 +/- sqrt(3)
- make a = 1 first
Important formulas
- add (half of b)^2
- then square root
- x = 2 +/- sqrt(3)
Common exam traps
- If a is not 1, divide the whole equation by a first, then complete the square.
- Not +6 or +16
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
If a is not 1, divide the whole equation by a first, then complete the square.
- Mistake
Not +6 or +16
- Mistake
Do not treat THE NUMBER TO ADD and SOLVE as the same step; each has a different purpose.
- Mistake
If a is not 1, divide the whole equation by a first, then complete the square.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
add (half of b)^2
then square root
x = 2 +/- sqrt(3)
make a = 1 first
For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is b
x = 2 +/- sqrt(3)
- Definition
- For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is b
- Formula
- x = 2 +/- sqrt(3)
- Example
- x^2+6x -> +9
- Critical warning
- If a is not 1, divide the whole equation by a first, then complete the square.
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Exam Practice
Define
Q1. Define Completing the Square.
Explain
Q2. Explain: THE NUMBER TO ADD.
Example
Q3. Give one clear example of Completing the Square and say why it fits.
Mistake
Q4. State one common mistake students make about Completing the Square.
Exam note
Q5. Write a short exam note on Completing the Square: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.