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Corbett MathsExpanding double brackets and factorising
Corbett MathsCompleting the square
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Key facts & methods
Expanding and factorising
- Double brackets: FOIL. (x+3)(x−2) = x²−2x+3x−6 = x²+x−6.
- Factorising x²+bx+c: find two numbers that multiply to c and add to b.
- E.g. x²+5x+6: need ×6 and +5 → 2 and 3 → (x+2)(x+3).
- Difference of two squares: a²−b² = (a+b)(a−b). E.g. x²−9 = (x+3)(x−3).
- Factorising ax²+bx+c: find two numbers that multiply to ac and add to b, then split and factor by grouping.
Completing the square
- x²+bx+c = (x + b/2)² − (b/2)² + c.
- E.g. x²+6x+2 = (x+3)² − 9 + 2 = (x+3)² − 7.
- For ax²+bx+c: factor out a first. E.g. 2x²+8x+1 = 2(x²+4x)+1 = 2(x+2)²−8+1 = 2(x+2)²−7.
- Uses: finding minimum/maximum point of a quadratic, solving quadratics.
- Minimum of y=(x+3)²−7 is at x=−3, y=−7.
Algebraic fractions
- Simplify: factorise top and bottom, then cancel common factors.
- E.g. (x²−4)/(x+2) = (x+2)(x−2)/(x+2) = x−2.
- Add/subtract: find common denominator (same process as ordinary fractions).
- E.g. 3/x + 2/(x+1) = 3(x+1)/[x(x+1)] + 2x/[x(x+1)] = (5x+3)/[x(x+1)].
- Multiply: multiply numerators together, denominators together, then simplify.
- Divide: keep, change, flip, then simplify.
Exam questions — 5 questions · 12 marks · Edexcel 4MA1 style
⚡ Show ALL working — the AI awards method marks just like a real examiner. Partial credit for correct working even if the final answer is wrong.
4 marksExpand and factorise4MA1 Paper style ✏️ No calculator
(a) Expand and simplify (2x+3)(x−4). (b) Factorise fully 6x²+x−12. (4 marks)
Hint: (a) FOIL. (b) For ax²+bx+c: find two numbers ×(ac) and +(b). Here ac=−72, b=1 → 9 and −8. Split the middle term then factor by grouping.
+40 XP
3 marksComplete the square4MA1 style ✏️ No calculator
Write x²−10x+18 in the form (x+a)²+b where a and b are integers. State the minimum value of x²−10x+18 and the value of x at which it occurs. (3 marks)
Hint: Half the coefficient of x: b/2 = −10/2 = −5. Write (x−5)², then subtract (−5)²=25 and add the constant 18.
+30 XP
3 marksAlgebraic fractionsGrade 7 Booklet style ✏️ No calculator
Simplify fully: (x²−9)/(x²+x−6). (3 marks)
Hint: Factorise both the numerator and denominator. Look for common factors to cancel. Top: difference of two squares. Bottom: standard factorisation.
+30 XP
1 markDifference of squares4MA1 style
Factorise completely: 4x² − 25
A (2x−5)²
B (4x+5)(x−5)
C (2x+5)(2x−5)
D (2x−25)
1 markCompleting the square — vertex4MA1 style
The quadratic y = (x−4)² + 3 has a minimum point. What are the coordinates of this minimum?
A (4, 3)
B (−4, 3)
C (4, −3)
D (−4, −3)
Module complete! 🎉
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