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Algebra

Algebra

Completing the square

Completed Square Form Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. The completed square form
  4. Turning point and minimum value
  5. Solving by completing the square
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 2.4 and 2.5: completing the square
  • 2.5Write \(x^{2}+bx+c\) (and \(ax^{2}+bx+c\)) in the form \((x+p)^{2}+q\).
  • 2.5Read off the turning point and the minimum (or maximum) value.
  • 2.5Solve a quadratic by completing the square, with exact (surd) answers.

Before you start: expanding squares and surds. Next: The quadratic formula.

Formula bank

Completing the squareLearn it\(x^{2}+bx+c=\left(x+\tfrac b2\right)^{2}-\left(\tfrac b2\right)^{2}+c\)halve b for the bracket, then subtract that number squared.Use it when 'write in the form \((x+p)^{2}+q\)'.
Turning pointLearn it\(y=(x+p)^{2}+q\) has turning point \((-p,\ q)\)minimum value \(q\) at \(x=-p\) (maximum if the \(x^{2}\) is negative).Use it when 'minimum value', 'turning point'.

The completed square form

\((x+3)^{2}=x^{2}+6x+9\). So \(x^{2}+6x\) is \((x+3)^{2}\) with 9 too many: \(x^{2}+6x=(x+3)^{2}-9\). The number in the bracket is always half of b.

General methodHow to write \(x^{2}+bx+c\) as \((x+p)^{2}+q\)

  1. Halve b: that is p. Write \((x+p)^{2}\).
  2. Subtract \(p^{2}\) (the extra bit the bracket makes).
  3. Add c and simplify: q = c − p².
  4. If there is a number \(a\) in front of \(x^{2}\): take \(a\) out of the \(x\) terms first, complete the square inside, then multiply back.

Example Write \(x^{2}-8x+3\) as \((x+p)^{2}+q\).

Half of −8 is −4; subtract 16; add 3.

  1. \((x-4)^{2}-16+3\)

Answer \((x-4)^{2}-13\)

Example 2 Write \(2x^{2}+12x+5\) as \(a(x+p)^{2}+q\).

Take 2 out of the \(x\) terms first.

  1. \(2(x^{2}+6x)+5\)
  2. \(2[(x+3)^{2}-9]+5\)
  3. \(2(x+3)^{2}-18+5\)

Answer \(2(x+3)^{2}-13\)

Try it yourself

1
\(x^{2}+6x+2=(x+p)^{2}+q\). Find p and q.
Show answer

Answer \(p=3,\ q=-7\)

2
\(x^{2}-10x=(x+p)^{2}+q\). Find p and q.
Show answer

Answer \(p=-5,\ q=-25\)

3
\(x^{2}+3x-1=(x+p)^{2}+q\). Find p and q.
Hint
Half of 3 is \(\tfrac32\); \(\left(\tfrac32\right)^{2}=\tfrac94\).
Show answer

Answer \(p=\tfrac32,\ q=-\tfrac{13}4\)

Turning point and minimum value

A square is never negative, so \((x+p)^{2}+q\) is smallest when the bracket is 0, at \(x=-p\). The smallest value is \(q\). On the graph the turning point is \((-p,\ q)\). With \(-(x-p)^{2}+q\) it is a maximum instead.

Completed squareTurning pointValue
\((x+3)^{2}-7\)\((-3,\ -7)\)minimum −7
\((x-4)^{2}-13\)\((4,\ -13)\)minimum −13
\(-(x-3)^{2}+7\)\((3,\ 7)\)maximum 7
The x-coordinate has the OPPOSITE sign to the number in the bracket: \((x+3)^{2}\) is smallest at \(x=-3\).

Solving by completing the square

General methodHow to solve by completing the square

  1. Complete the square: \((x+p)^{2}+q=0\).
  2. Move q across: \((x+p)^{2}=-q\).
  3. Square-root both sides, with ±.
  4. Subtract p. Leave surds if they ask for exact answers.

Example Solve \(x^{2}+10x+6=0\), leaving your answers in surd form.

Complete the square, then square-root with ±.

  1. \((x+5)^{2}-25+6=0\)
  2. \((x+5)^{2}=19\)
  3. \(x+5=\pm\sqrt{19}\)

Answer \(x=-5\pm\sqrt{19}\)

Try it yourself

1
What is the minimum value of \(x^{2}+4x+9\)?
Hint
\((x+2)^{2}+5\).
Show answer

Answer 5

2
Solve \(x^{2}-14x+40=0\) by completing the square.
Hint
\((x-7)^{2}=9\).
Show answer

Answer 4, 10

Ways they test it

Type 1 Write in the form

"Write \(x^{2}-7x+5\) in the form \((x-a)^{2}-b\)." (2 marks)

Spot it the form is given with letters.

  1. Halve b.
  2. Subtract its square, add c.

For example \(x^{2}-7x+5\)

  1. \(\left(x-\tfrac72\right)^{2}-\tfrac{49}4+5\)

Answer \(\left(x-\tfrac72\right)^{2}-\tfrac{29}4\)

Type 2 Hence the minimum

"Hence write down the minimum value / the turning point." (1 mark)

Spot it 'hence' after the form.

  1. Minimum value = q.
  2. Turning point = \((-p,\ q)\).

Type 3 Hence solve, exact answers

"Hence solve \(x^{2}+8x+3=0\), giving your answers in the form \(p\pm\sqrt q\)." (2 marks)

Spot it 'exact' or 'surd form'.

  1. \((x+p)^{2}=\ldots\)
  2. ± root, then subtract p.

For example \((x+4)^{2}=13\)

    Answer \(-4\pm\sqrt{13}\)

    Type 4 Number in front of x²

    "Write \(3x^{2}-12x+5\) in the form \(a(x+p)^{2}+q\)." (3 marks, 4024)

    Spot it a coefficient on \(x^{2}\).

    1. Take a out of the x terms.
    2. Complete the square inside.
    3. Multiply back and simplify.

    For example \(3(x^{2}-4x)+5\)

    1. \(3[(x-2)^{2}-4]+5\)

    Answer \(3(x-2)^{2}-7\)

    Mistakes to avoid

    • ✗ \(x^{2}+6x+2=(x+6)^{2}+\ldots\)✓ \((x+3)^{2}-7\)Halve b.
    • ✗ \((x+3)^{2}+9+2\)✓ \((x+3)^{2}-9+2\)SUBTRACT the square.
    • ✗ \((x+3)^{2}-7\): minimum at \(x=3\)✓ at \(x=-3\)Opposite sign.
    • ✗ \((x+5)^{2}=19\Rightarrow x=-5+\sqrt{19}\)✓ \(x=-5\pm\sqrt{19}\)Two square roots.

    Summary

    • \(x^{2}+bx+c=(x+\tfrac b2)^{2}-(\tfrac b2)^{2}+c\).
    • Turning point \((-p,\ q)\); minimum value q.
    • Solving: isolate the square, ± root.
    • Number in front of \(x^{2}\): take it out of the x terms first.

    If you see …

    • 'in the form \((x+p)^{2}+q\)'halve b, subtract its square.
    • 'exact answers' / 'surd form'complete the square and ± root.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    \(x^{2}+4x+9=(x+p)^{2}+q\). Find p and q.
    Show answer

    Answer \(p=2,\ q=5\)

    2
    \(x^{2}-8x+3=(x+p)^{2}+q\). Find p and q.
    Show answer

    Answer \(p=-4,\ q=-13\)

    3
    Write down the turning point of \(y=(x-4)^{2}-13\).
    Show answer

    Answer \((4,\ -13)\)

    Medium

    1
    \(x^{2}-6x+11=(x-a)^{2}+b\). Find a and b, and the minimum value.
    Guide me

    What is it asking? a, b, minimum

    What do we know? half of −6 is −3

    Plan subtract 9, add 11

    Show working
    1. \((x-3)^{2}-9+11=(x-3)^{2}+2\)
    2. minimum value 2 (at x = 3)

    Answer \(a=3,\ b=2\); minimum 2

    2
    Solve \(x^{2}+8x+3=0\) by completing the square. Give exact answers.
    Guide me

    What is it asking? x, exact

    What do we know? \((x+4)^{2}-16+3\)

    Plan \((x+4)^{2}=13\)

    Show working
    1. \((x+4)^{2}=13\)
    2. \(x=-4\pm\sqrt{13}\)

    Answer \(-4\pm\sqrt{13}\)

    3
    \(-x^{2}+6x-2=-(x-p)^{2}+q\). Find p and q, and the maximum value.
    Guide me

    What is it asking? p, q

    What do we know? take −1 out: \(-(x^{2}-6x)-2\)

    Plan \(-[(x-3)^{2}-9]-2\)

    Show working
    1. \(-(x-3)^{2}+9-2\)
    2. \(-(x-3)^{2}+7\)

    Answer \(p=3,\ q=7\); max 7

    Hard

    1
    \(3x^{2}-12x+5=a(x+p)^{2}+q\). Find a, p and q.
    Guide me

    What is it asking? a, p, q

    What do we know? take 3 out of the x terms

    Plan \(3[(x-2)^{2}-4]+5\)

    Show working
    1. \(3(x^{2}-4x)+5\)
    2. \(3(x-2)^{2}-12+5\)
    3. \(3(x-2)^{2}-7\)

    Answer \(a=3,\ p=-2,\ q=-7\)

    PartAnswer / workingMarksGuidance
    a = 3B1
    p = −2B1
    q = −7B1
    Total: 3 marks
    2
    \(x^{2}-7x+5=(x-a)^{2}-b\). Find a and b.
    Guide me

    What is it asking? a, b

    What do we know? half of −7 is \(-\tfrac72\)

    Plan subtract \(\tfrac{49}4\), add 5

    Show working
    1. \(\left(x-\tfrac72\right)^{2}-\tfrac{49}4+5\)
    2. \(\left(x-\tfrac72\right)^{2}-\tfrac{29}4\)

    Answer \(a=\tfrac72,\ b=\tfrac{29}4\)

    PartAnswer / workingMarksGuidance
    \(a=\tfrac72\)B1
    \(b=\tfrac{29}4\)B1
    Total: 2 marks
    3
    Solve \(x^{2}-12x+30=0\), giving your answers in the form \(p\pm\sqrt q\).
    Guide me

    What is it asking? x, exact

    What do we know? \((x-6)^{2}-36+30\)

    Plan \((x-6)^{2}=6\)

    Show working
    1. \((x-6)^{2}=6\)
    2. \(x=6\pm\sqrt6\)

    Answer \(6\pm\sqrt6\)

    PartAnswer / workingMarksGuidance
    \((x-6)^{2}-6\)M1
    \(6\pm\sqrt6\)A1
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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