You can solve quadratic equations that don’t have integer
(whole number) answers by completing the square. Completed Square Form: (x + p)** ^{2}**
+ q

Useful identities when completing the square, these will allow you to save time:

X** ^{2}** + 2bx + c = (x + b)

**– b + c**

^{2}X** ^{2}** – 2bx + c = (x – b)

**– b + c**

^{2}It is important to remember that when square rooting a positive number, it will have two square roots, one being positive and one being negative. If you ‘square root’ both sides of an equation in your working out, then remember to show the ‘plus-or-minus’ symbol (±). This shows that there are two square roots, for example:

x** ^{2}** = 4 then x = ± 2

Or

(x + 4) )** ^{2}** = 3

(x + 4) = √3

x = -4 ± √3

Worked example of completing the square:

a) find values of p and q such that x** ^{2}** + 6x – 20 ≡ (x + p)

**+ q (2marks)**

^{2}b) Hence or otherwise, solve the equation:

x** ^{2}** + 6x – 20

Give your answer in surd form: (2marks)

c) Write 4x** ^{2}** + 8x – 11 in the form a(x + p)

**+ p (2 marks)**

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