NCEA Level 2Quadratic equationsAS 91261

Quadratic equations: solving, and forming from the roots

What this lesson covers

  • Every quadratic can be solved by factorising, the formula, or completing the square, and each suits different equations.
  • The roots of a quadratic fix its factors, so the equation can be rebuilt from its solutions.
Key idea

A quadratic equation

  • ax² + bx + c = 0.
  • Usually two solutions, called roots.
  • Three ways: factorise, formula, or graph.
Worked example

Roots are where the graph crosses

  • y = x² − 5x − 14.
  • It crosses the x-axis at −2 and 7.
  • Those are the solutions of x² − 5x − 14 = 0.
Roots are where the graph crosses
Worked example

Solving by factorising

  • Product −14, sum −5.
  • A factor must be zero.
  1. x² − 5x − 14 = 0
  2. −7 × 2 = −14, −7 + 2 = −5
  3. (x − 7)(x + 2) = 0factorise
  4. x − 7 = 0 or x + 2 = 0a factor is zero
  5. x = 7 or x = −2
Worked example

Rearrange to zero first

  • Make one side zero.
  • Then factorise.
  1. x(x + 3) = 10
  2. x² + 3x = 10expand
  3. x² + 3x − 10 = 0subtract 10
  4. (x + 5)(x − 2) = 05 × (−2) = −10
  5. x = −5 or x = 2
  6. x = 2: 2 × 5 = 10
Worked example

When it will not factorise

  • Write a, b, c with signs.
  • Brackets round negatives.
  1. x = (−b ± √(b² − 4ac))/2a
  2. 2x² − 3x − 4 = 0
  3. a = 2, b = −3, c = −4
  4. b² − 4ac = (−3)² − 4(2)(−4)
  5. = 9 + 32 = 41
  6. x = (3 ± √41)/4−b = 3, 2a = 4
  7. x = (3 + √41)/4 = 2.351
  8. x = (3 − √41)/4 = −0.851
Worked example

Working backwards: roots to equation

  • Each root gives a factor.
  • Clear the fraction first.
  1. x = 3, so x − 3 = 0
  2. x = −1/2
  3. 2x = −1, so 2x + 1 = 0double both sides
  4. (x − 3)(2x + 1) = 0
  5. 2x² + x − 6x − 3 = 0expand
  6. 2x² − 5x − 3 = 0collect
Key idea

What a fractional root tells you

  • A root x = p/q comes from a factor (qx − p).
  • So the leading coefficient is a multiple of q.
  • Fractional roots mean the equation factorises.
Worked example

In context: reject what cannot be

  • Write the equation first.
  • Reject what cannot be.
  1. x(x + 3) = 40width × length
  2. x² + 3x − 40 = 0expand, subtract 40
  3. (x + 8)(x − 5) = 08 × (−5) = −40
  4. x = −8 or x = 5
  5. x = −8a width is positive
  6. x = 5, length 5 + 3 = 8
  7. 5 m by 8 m
  8. 5 × 8 = 40 m²
Key idea

State the equation you formed

  • Write the equation before solving it.
  • The specification asks for this.
  • An answer alone is not enough.
Key idea

Where the marks are

  • Achieved: solve, and form from roots.
  • Merit: in context, choosing the method.
  • Excellence: modelling a real situation.
Summary

What to take away

  • Equal to zero first, then factorise or use the formula.
Firstrearrange to = 0
Factorisewhen the numbers are friendly
Formulax = (−b ± √(b² − 4ac)) / 2a
From rootseach root gives a factor
Fractional rootx = p/q gives (qx − p)
Contextreject impossible answers
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Question 3: whole-number coefficients.
  • Question 4 needs the formula.
Question
1Solve x² + 2x − 15 = 0.
2Solve 3x² + x − 2 = 0.
3Form a quadratic with roots 2 and −3/4.
4Solve x² − 4x − 1 = 0, to 3 decimal places.
Show answer

1. (x + 5)(x − 3) = 0: x = 3 or −5. 2. (3x − 2)(x + 1) = 0: x = 2/3 or −1. 3. (x − 2)(4x + 3) = 4x² − 5x − 6 = 0. 4. x = (4 ± √20)/2 = 2 ± √5: 4.236 or −0.236.

Worked example

The answers: questions 1 and 2

  1. 1. x² + 2x − 15 = 0
  2. (x + 5)(x − 3) = 05 × (−3) = −15, 5 − 3 = 2
  3. x = −5 or x = 3
  4. 2. 3x² + x − 2 = 0
  5. a × c = −6: use 3 and −2
  6. 3x² + 3x − 2x − 2 = 0split x
  7. 3x(x + 1) − 2(x + 1) = 0group in pairs
  8. (3x − 2)(x + 1) = 0common bracket
  9. x = 2/3 or x = −13x = 2
Worked example

The answers: questions 3 and 4

  1. 3. x = 2 gives x − 2
  2. x = −3/4 gives 4x + 34x = −3
  3. (x − 2)(4x + 3) = 0
  4. 4x² + 3x − 8x − 6 = 0expand
  5. 4x² − 5x − 6 = 0collect
  6. 4. b² − 4ac = (−4)² − 4(1)(−1)
  7. = 16 + 4 = 20
  8. x = (4 ± √20)/2 = 2 ± √5√20 = 2√5
  9. x = 4.236 or x = −0.2363 decimal places

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