NCEA Level 2Algebraic fractionsAS 91261

Algebraic fractions: 7 exam questions solved step by step

What this lesson covers

  • A fraction can only be cancelled by a factor of the whole numerator and the whole denominator.
  • Factorising first turns a hard fraction into one where the common factor can be seen.
Key idea

7 questions, one habit

  • 1. Expand every term by every term.
  • 2. Factorise fully: common factor first.
  • 3. Cancel only whole factors.
  • 4. Never let a denominator be zero.
Your turn

Question 1 · Achieved

  • Expand and simplify
  • (4x − 3)(2x + 5) − 2x(x − 1)
Worked example

Q1: expand and simplify

  • Every term by every term.
  • The minus goes to both.
  1. (4x − 3)(2x + 5)
  2. = 8x² + 20x − 6x − 15four products
  3. = 8x² + 14x − 15collect
  4. −2x(x − 1) = −2x² + 2x−2x × −1 = +2x
  5. 8x² + 14x − 15 − 2x² + 2x
  6. = 6x² + 16x − 15collect like terms
  7. −2x² − 2x−2x × −1 is +2x
Your turn

Question 2 · Achieved

  • Factorise fully 3x² − 12x − 36
Worked example

Q2: 3x² − 12x − 36

  • Common factor first.
  • "Fully" means nothing left.
  1. 3x² − 12x − 36 = 3(x² − 4x − 12)
  2. −6 × 2 = −12, −6 + 2 = −4
  3. = 3(x − 6)(x + 2)factorise the bracket
  4. 3(x² − 6x + 2x − 12)expand back
  5. = 3x² − 12x − 36
  6. (3x + 6)(x − 6)3 is still a factor
Your turn

Question 3 · Merit

  • Simplify (6x² − 7x − 20)/(4x² − 25)
Worked example

Q3: factorise top and bottom

  • Top: a is 6, use ac.
  • Bottom: two squares.
  1. top: ac = 6 × −20 = −120
  2. −15 × 8 = −120, −15 + 8 = −7
  3. 6x² − 15x + 8x − 20split −7x
  4. = 3x(2x − 5) + 4(2x − 5)group in pairs
  5. = (3x + 4)(2x − 5)common bracket
  6. 4x² − 25 = (2x − 5)(2x + 5)two squares
Worked example

Q3: cancel whole factors

  • Only whole factors cancel.
  • Check with a number.
  1. top = (3x + 4)(2x − 5)
  2. bottom = (2x − 5)(2x + 5)
  3. (2x − 5) is a factor of both
  4. = (3x + 4)/(2x + 5)cancel (2x − 5)
  5. x ≠ 5/2, x ≠ −5/2bottom cannot be 0
  6. 6x²/4x² and −20/−25terms, not factors
  7. x = 0: −20/−25 = 4/5
  8. (0 + 4)/(0 + 5) = 4/5
Your turn

Question 4 · Merit

  • Write as a single fraction
  • x/(x² − 9) + 2/(x + 3)
Worked example

Q4: x/(x² − 9) + 2/(x + 3)

  • Factorise the denominator.
  • Use the lowest one.
  1. x² − 9 = (x − 3)(x + 3)two squares
  2. LCD = (x − 3)(x + 3)holds x + 3 already
  3. top: x + 2(x − 3)second top × (x − 3)
  4. = x + 2x − 6expand
  5. = 3x − 6 = 3(x − 2)collect, factorise
  6. = 3(x − 2)/((x − 3)(x + 3))
  7. x = 4: 4/7 + 2/7 = 6/7
  8. 3(2)/(1 × 7) = 6/7
Your turn

Question 5 · Merit, in context

  • Aroha cycles 20 km along a trail at x km/h,
  • then rides back 5 km/h faster.
  • (a) Write her total time as one fraction.
  • (b) Find her total time when x = 20.
Worked example

Q5 (a): total time

  • time = distance ÷ speed
  • One fraction: one LCD.
  1. T = 20/x + 20/(x + 5)time = distance ÷ speed
  2. LCD = x(x + 5)
  3. T = (20(x + 5) + 20x)/(x(x + 5))
  4. = (20x + 100 + 20x)/(x(x + 5))
  5. = (40x + 100)/(x(x + 5))collect
  6. = 20(2x + 5)/(x(x + 5))factor out 20
Worked example

Q5 (b): when x = 20

  • Substitute, then check.
  • Answer in words.
  1. T = 20(2x + 5)/(x(x + 5))
  2. x = 20: T = 20 × 45/(20 × 25)
  3. = 900/500 = 1.8 h
  4. 20/20 + 20/25 = 1 + 0.8
  5. 0.8 h = 0.8 × 60 = 48 min
  6. Total time 1 hour 48 minutes
Your turn

Question 6 · Excellence

  • Simplify fully
  • (2x² + x − 6)/(x² − 9) ÷ (4x² − 9)/(x² − 3x)
Worked example

Q6: flip, then factorise all four

  • Divide: flip and multiply.
  • Factorise every part.
  1. × (x² − 3x)/(4x² − 9)flip the second
  2. 2x² + x − 6: ac = −12, 4 and −3
  3. = 2x² + 4x − 3x − 6split x
  4. = 2x(x + 2) − 3(x + 2)group in pairs
  5. = (2x − 3)(x + 2)common bracket
  6. x² − 9 = (x − 3)(x + 3)two squares
  7. 4x² − 9 = (2x − 3)(2x + 3)two squares
  8. x² − 3x = x(x − 3)common factor x
Worked example

Q6: multiply and cancel

  • Cancel whole factors.
  • Check with x = 1.
  1. (2x − 3)(x + 2)/((x − 3)(x + 3))
  2. × x(x − 3)/((2x − 3)(2x + 3))
  3. cancel (2x − 3) and (x − 3)
  4. = x(x + 2)/((x + 3)(2x + 3))
  5. x = 1: (3/8) ÷ (5/2) = 3/20
  6. 1 × 3/(4 × 5) = 3/20
Your turn

Question 7 · Excellence

  • Solve x/(x − 2) + 1/(x + 2) = 8/(x² − 4)
Worked example

Q7: clear the fractions

  • Note what x cannot be.
  • Multiply every term.
  1. x² − 4 = (x − 2)(x + 2)two squares
  2. x ≠ 2, x ≠ −2no zero denominators
  3. × (x − 2)(x + 2), every term
  4. x(x + 2) + (x − 2) = 8denominators cancel
  5. x² + 2x + x − 2 = 8expand
  6. x² + 3x − 10 = 0collect, −8
Worked example

Q7: solve, then reject

  • Test each root against x ≠ ±2.
  • Check in the original.
  1. x² + 3x − 10 = 0
  2. x ≠ 2, x ≠ −2
  3. 5 × −2 = −10, 5 + −2 = 3
  4. (x + 5)(x − 2) = 0factorise
  5. x = −5 or x = 2
  6. x = 2makes x − 2 = 0
  7. x = −5: 5/7 − 1/3 = 8/21
  8. 8/(25 − 4) = 8/21
  9. x = −5 is the only solution
Summary

The habits that solved all seven

  • Factorise first, cancel factors, watch the denominator.
StepHow
Expandevery term by every term
Factorisecommon factor first, then ac
Cancelonly whole factors
Addfactorise, then lowest LCD
Divideflip the second, multiply
Solvereject a zero denominator

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