Ws Nd1 Quad Eq _1_ Products _1_ B
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Quadratic Expressions (Binomial Products) Important Formulae -
-
(a ± b)2 = a2 ± 2ab + b2 2 ( ± )
(a + b)(a (
+
b) = a2
=
2
± 2
+
b2
) (
)
=
2
2
EXERCISE 1 1. Use the diagram to prove: (a) (a + b) 2 = a 2 + 2ab + b 2
(b) ( x + p )2 = x 2 + 2 px + p 2
(c) (a b)2 = a 2 2ab + b 2
(d) ( x
(e) (a + b)(a b) = a 2 b 2
(f) ( x + p )( x
p ) 2 = x 2 2 px + p 2
p) = x 2
p2
2
2. Use the formula to expand: (a) ( x + 5)2 = (
)2
+
(b) ( x + 10) 2 = (
Ans: x 2 + 10 x + 25 )2
(c) (2 x 5)2 = (
(d) (2 x 3) 2 = (
)2
+
(g)
(f) (3 x 2 y ) 2
+
)
2
(h)
Ans: (i)
2x +
Ans: 9 x 2 12 xy + 4 y 2
2
=(
3y 4
)2
Ans: 4 x 2 12 x + 9
Ans: 4 x 2 + 12 xy + 9 y 2 x +y 2
)2
Ans: x 2 + 20 x + 100
Ans: 4 x 2 20 x + 25 (e) (2 x + 3 y ) 2 = (
+
x 2
2
x2 + xy + y 2 4
Ans:
2
(i)
3 xy 9 y 2 Ans: 4 x + + 4 16 2
y 3
5x2
4y 3
x 2 xy y 2 + + 4 3 9
2
40 x 2 y 16 y 2 Ans: 25 x + + 3 9 4
3. Use the formula to expand: (a) ( x + 2)( x 2) = (
+
)(
+
(b) ( x + 5)( x 5) = (
)
+
)(
+
Ans: x 2 4 (c) (2 x + 3)(2 x 3) = (
+
)(
+
)
Ans: x 2 25 (d) (2 x + 3)(2 x 3) = (
Ans: 4 x 2 9 (e) (2 x + 3 y )(2 x 3 y )
x +y 2
x 2
2x
3y 4
+
)
Ans: 9 x 2 4 y 2 (h)
y
2x +
)(
(f) (3 x + 2 y )(3 x 2 y )
Ans: (i)
+
Ans: 4 x 2 9
Ans: 4 x 2 9 y 2 (g)
)
x2 4
x 2
y 3
y2
3y 4
Ans: (j)
Ans: 4 x 2
x y + 2 3
9 y2 16
5x2
4y 3
5x2 +
x2 4
y2 9
4y 3
Ans: 25 x 4
16 y 2 9
(k) ( x y )( x + y )( x 2 + y 2 )( x 4 + y 4 )
Ans: x8
y8
-
(a ± b)2 = a2 ± 2ab + b2 2 ( ± )
(a + b)(a (
+
b) = a2
=
2
± 2
+
b2
) (
)
=
2
2
EXERCISE 1 1. Use the diagram to prove: (a) (a + b) 2 = a 2 + 2ab + b 2
(b) ( x + p )2 = x 2 + 2 px + p 2
(c) (a b)2 = a 2 2ab + b 2
(d) ( x
(e) (a + b)(a b) = a 2 b 2
(f) ( x + p )( x
p ) 2 = x 2 2 px + p 2
p) = x 2
p2
2
2. Use the formula to expand: (a) ( x + 5)2 = (
)2
+
(b) ( x + 10) 2 = (
Ans: x 2 + 10 x + 25 )2
(c) (2 x 5)2 = (
(d) (2 x 3) 2 = (
)2
+
(g)
(f) (3 x 2 y ) 2
+
)
2
(h)
Ans: (i)
2x +
Ans: 9 x 2 12 xy + 4 y 2
2
=(
3y 4
)2
Ans: 4 x 2 12 x + 9
Ans: 4 x 2 + 12 xy + 9 y 2 x +y 2
)2
Ans: x 2 + 20 x + 100
Ans: 4 x 2 20 x + 25 (e) (2 x + 3 y ) 2 = (
+
x 2
2
x2 + xy + y 2 4
Ans:
2
(i)
3 xy 9 y 2 Ans: 4 x + + 4 16 2
y 3
5x2
4y 3
x 2 xy y 2 + + 4 3 9
2
40 x 2 y 16 y 2 Ans: 25 x + + 3 9 4
3. Use the formula to expand: (a) ( x + 2)( x 2) = (
+
)(
+
(b) ( x + 5)( x 5) = (
)
+
)(
+
Ans: x 2 4 (c) (2 x + 3)(2 x 3) = (
+
)(
+
)
Ans: x 2 25 (d) (2 x + 3)(2 x 3) = (
Ans: 4 x 2 9 (e) (2 x + 3 y )(2 x 3 y )
x +y 2
x 2
2x
3y 4
+
)
Ans: 9 x 2 4 y 2 (h)
y
2x +
)(
(f) (3 x + 2 y )(3 x 2 y )
Ans: (i)
+
Ans: 4 x 2 9
Ans: 4 x 2 9 y 2 (g)
)
x2 4
x 2
y 3
y2
3y 4
Ans: (j)
Ans: 4 x 2
x y + 2 3
9 y2 16
5x2
4y 3
5x2 +
x2 4
y2 9
4y 3
Ans: 25 x 4
16 y 2 9
(k) ( x y )( x + y )( x 2 + y 2 )( x 4 + y 4 )
Ans: x8
y8
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