Category Archives: Year 9 Mathematics

November 11, 2025 · 10:19 am

Completing the Square

x^2+6x-4 \rightarrow (x+3)^2-13

Completing the square is useful to

  • sketch parabolas.
  • solve quadratics.
  • factorising quadratics
  • finding the centre and radius version of the equation of a circle.

When completing the square we take advantage of perfect squares. For example, (x+3)^2=(x+3)(x+3)=x^2+6x+9

6=2\times 3 and 9=3\times 3

Example 1

Put x^2+8x-5 into completed square form.

What perfect square has an 8x term?

(x+4)^2=x^2+8x+16

We don’t want +16, we want -5, so subtract 16+5

x^2+8x-5=(x+4)^2-21

x^2+bx+c=(x+\frac{b}{2})^2-(\frac{b}{2})^2+c

What about a non-monic quadratic? For example,

2x^2+12x+11

Factorise the 2

2(x^2+6x+\frac{11}{2})

And continue as before

2[(x+3)^2-9+\frac{11}{2}]=2[(x+3)^2-\frac{18}{2}+\frac{11}{2}]=2[(x+3)^2-\frac{7}{2}]=2(x+3)^2-7

Example 2

y=2x^2+7x-5

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

2[(x+\frac{7}{4})^2-(\frac{7}{4})^2-\frac{5}{2}]

2[(x+\frac{7}{4})^2-\frac{49}{16}-\frac{40}{16}]

2[(x+\frac{7}{4})^2-\frac{89}{16}]

2(x+\frac{7}{4})^2-\frac{89}{8}

ax^2+bx+c=a(x+\frac{b}{2a})^2-a((\frac{b}{2a})^2+\frac{c}{a})

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Filed under Algebra, Arithmetic, Classpad Skills, Completing the Square, Fractions, Quadratic, Quadratics, Year 10 Mathematics, Year 9 Mathematics

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