Quadratic Expressions Worksheets Page 21
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36number and algebra
WOrKed eXamPle 10
WOrKed eXamPle 10
WOrKed eXamPle 10
Write the following in turning point form by completing the square.
+ 4x
2
x
a
+ 7x + 1
2
x
b
THInK
WrITe
a
• The square will consist of a square
a
+
x
2
2
that has an area of x
and two identical
rectangles with a total area of 4x.
2
x
x
2x
x
• The length of the large square is (x + 2)
so its area is (x + 2)
2
.
+
2
2x
• The area of the smaller square is (2)
.
2
+ 4x in turning point form.
x
2
• Write x
2
2
2
+ 4x = (x + 2)
2
− (2)
2
x
= (x + 2)
− 4
2
b
1
• Complete the square with the terms
b
7
+
x
2
containing x.
• The square will consist of a square
+ 7x + 1 = a x +
7
b
− a
b
x
x
x
2
x
2
that has an area of x
and two
2
ax +
b
ax +
b
identical rectangles with a total area
+
( )
7
7
7
2
of 7x.
x
2
2
2
• The length of the large square is
x
a
b
2
2
2
7
7
+ 1
2
x
7
7
so its area is
.
2
2
2
2
• The area of the smaller square is
2
7
= a x +
b
.
2
+ 7x + 1 in turning point
2
• Write x
= a x +
b
form.
2
7
49
4
−
+
2
Simplify the last two terms.
2
4
4
2
7
45
−
2
4
+ a
b
− a
b
• The process of completing the square is sometimes described as the process of adding
the square of half of the coeffi cient of x then subtracting it, as shown in green below. The
+ bx + a
b
a
b
result of this process is a perfect square that is then factorised, as shown in blue.
2
2
b
b
+ bx = x
+ bx
2
2
x
a x +
b
a
b
2
2
2
2
b
b
=
−
2
x
2
2
2
2
b
b
=
−
2
2
Maths Quest 10 + 10A
288
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