NCEA Level 1Quadratic equationsAS 91947

Quadratic equations: 7 exam questions solved step by step

What this lesson covers

  • If two quantities multiply to zero, at least one of them is zero.
  • A quadratic equation can have two, one or no real solutions, and the solutions are where its graph crosses the x-axis.
Key idea

7 questions, one method

  • 1. Rearrange so one side is 0.
  • 2. Factorise.
  • 3. Each bracket = 0 (null factor law).
  • 4. Check every answer.
Your turn

Question 1 · Achieved

  • Solve x² + 2x − 15 = 0
Worked example

Q1: x² + 2x − 15 = 0

  • Multiply to −15, add to 2.
  • Check both roots.
  1. 5 × −3 = −15, 5 + −3 = 2
  2. (x + 5)(x − 3) = 0factorise
  3. x + 5 = 0 or x − 3 = 0null factor law
  4. x = −5
  5. x = 3
  6. x = 3: 9 + 6 − 15 = 0
  7. x = −5: 25 − 10 − 15 = 0
  8. x = −5 or x = 3
Your turn

Question 2 · Achieved

  • Solve x² = 7x − 10
Worked example

Q2: x² = 7x − 10

  • Not 0 yet: rearrange.
  • Check in the original.
  1. x² = 7x − 10
  2. x² − 7x + 10 = 0−7x, +10 both sides
  3. −2 × −5 = 10, −2 + −5 = −7
  4. (x − 2)(x − 5) = 0factorise
  5. x = 2 or x = 5null factor law
  6. x = 2: 4 = 14 − 10
  7. x = 5: 25 = 35 − 10
Your turn

Question 3 · Merit

  • Solve 3x² − 10x − 8 = 0
Worked example

Q3: 3x² − 10x − 8 = 0

  • a = 3: use ac.
  • Split the middle term.
  1. ac = 3 × −8 = −24
  2. −12 × 2 = −24, −12 + 2 = −10
  3. 3x² − 12x + 2x − 8 = 0split the middle
  4. 3x(x − 4) + 2(x − 4) = 0factorise in pairs
  5. (3x + 2)(x − 4) = 0common bracket
  6. 3x = −2, x = −2/3
  7. x = 4
  8. x = 4: 48 − 40 − 8 = 0
  9. x = −2/3 or x = 4
Your turn

Question 4 · Merit

  • Solve (x + 2)(x − 3) = 14
  • Careful: the right side is 14, not 0.
Worked example

Q4: (x + 2)(x − 3) = 14

  • Brackets = 14 tells you nothing.
  • Expand, then get to 0.
  1. x² − 3x + 2x − 6 = 14expand
  2. x² − x − 6 = 14collect x terms
  3. x² − x − 20 = 0−14 both sides
  4. −5 × 4 = −20, −5 + 4 = −1
  5. (x − 5)(x + 4) = 0factorise
  6. x = 5 or x = −4null factor law
  7. x = 5: 7 × 2 = 14
  8. x = −4: (−2)(−7) = 14
Your turn

Question 5 · Merit, in context

  • A ball is thrown up.
  • Height after t seconds: h = 20t − 5t²
  • When is the ball 15 m high?
Worked example

Q5: when is h = 15?

  • Keep t² positive.
  • Two answers can both fit.
  1. 20t − 5t² = 15h = 15
  2. 0 = 5t² − 20t + 15all to the right
  3. 0 = t² − 4t + 3÷ 5
  4. 0 = (t − 1)(t − 3)−1 × −3, −1 + −3
  5. t = 1 or t = 3
  6. t = 1: 20 − 5 = 15
  7. t = 3: 60 − 45 = 15
  8. 1 s going up, 3 s coming down
Your turn

Question 6 · Excellence

  • Two consecutive odd numbers
  • multiply to 143.
  • Find the two numbers.
Worked example

Q6: odd numbers, product 143

  • Define the unknown.
  • Reject only with a reason.
  1. Let n = the smaller odd number
  2. next odd number = n + 2
  3. n(n + 2) = 143
  4. n² + 2n − 143 = 0expand, −143
  5. (n + 13)(n − 11) = 013 × −11, 13 + −11
  6. n = −13 or n = 11
  7. 11 × 13 = 143
  8. (−13)(−11) = 143
  9. 11 and 13, or −13 and −11
Your turn

Question 7 · Excellence

  • A right-angled triangle has sides
  • x, x + 7 and hypotenuse x + 8 (cm).
  • Find the three side lengths.
Question 7 · Excellence
Worked example

Q7: form the equation

  • Pythagoras: a² + b² = c²
  • Expand each square fully.
  1. x² + (x + 7)² = (x + 8)²Pythagoras
  2. (x + 7)² = x² + 14x + 49
  3. (x + 8)² = x² + 16x + 64
  4. x² + x² + 14x + 49
  5. = x² + 16x + 64
  6. 2x² + 14x + 49 = x² + 16x + 64
  7. x² − 2x − 15 = 0−x², −16x, −64
Worked example

Q7: solve, reject, check

  • A length cannot be negative.
  • Answer in words, with units.
  1. x² − 2x − 15 = 0
  2. (x − 5)(x + 3) = 0−5 × 3, −5 + 3
  3. x = 5 or x = −3
  4. x = −3side would be −3 cm
  5. sides 5, 12, 13x, x + 7, x + 8
  6. 25 + 144 = 169 = 13²
  7. 5 cm, 12 cm and 13 cm
Summary

The method that solved all seven

  • Zero, factorise, solve, check, and reason in context.
StepHow
Get to zeroeverything on one side first
Factorisemultiply to c, add to b
a is not 1split the middle using ac
Solveeach bracket equals 0
Checksubstitute both roots back
In contextreject only with a reason

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