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Solving Quadratics073 .. 2 xxge
Solving Quadratics073 .. 2 xxge
2193
22893
x
Solving Quadratics073 .. 2 xxge
2193
22893
x
2193 i
2193or
2193 ixix
Solving Quadratics073 .. 2 xxge
2193
22893
x
OR
0732 xx
2193 i
2193or
2193 ixix
by completing the square
Solving Quadratics073 .. 2 xxge
2193
22893
x
OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
by completing the square
Solving Quadratics073 .. 2 xxge
2193
22893
x
OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
0219
23
219
23
ixix
by completing the square
Solving Quadratics073 .. 2 xxge
2193
22893
x
OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
0219
23
219
23
ixix
ixix219
23or
219
23
by completing the square
Creating QuadraticsAll quadratics with real coefficients have
conjugate pairs as solutions
Creating QuadraticsAll quadratics with real coefficients have
conjugate pairs as solutions
2 ( )x z x z x z z x zz
Creating QuadraticsAll quadratics with real coefficients have
conjugate pairs as solutions
2 ( )x z x z x z z x zz
2 2Re( )x z x zz
Creating QuadraticsAll quadratics with real coefficients have
conjugate pairs as solutions
2 ( )x z x z x z z x zz
2 2Re( )x z x zz
-iiiii 4and4rootsith equation wquadratic theFind
Creating QuadraticsAll quadratics with real coefficients have
conjugate pairs as solutions
2 ( )x z x z x z z x zz
2 2Re( )x z x zz
8
-iiiii 4and4rootsith equation wquadratic theFind
17
Creating QuadraticsAll quadratics with real coefficients have
conjugate pairs as solutions
2 ( )x z x z x z z x zz
2 2Re( )x z x zz
8
0178 isequation 2 xx
-iiiii 4and4rootsith equation wquadratic theFind
17
Finding Square Rootsa ib x iy
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
1222 ba 162 ab
ab 8
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
1222 ba 162 ab
ab 81264
22
aa
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
1222 ba 162 ab
ab 81264
22
aa
06412 24 aa
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
1222 ba 162 ab
ab 81264
22
aa
06412 24 aa 0164 22 aa
16or 4 22 aa
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
1222 ba 162 ab
ab 81264
22
aa
06412 24 aa 0164 22 aa
16or 4 22 aa
42
ba solutionsreal no
Finding Square Rootsa ib x iy
2a ib x iy 2 22a abi b x iy
ibaiyx If
yabxba
2then 22
ie.g. 1612 Find
1222 ba 162 ab
ab 81264
22
aa
06412 24 aa 0164 22 aa
16or 4 22 aa
42
ba solutionsreal no ii 421612
ORa ib x iy
ORa ib x iy
2 2
2a b x
ab y
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
Note: y > 0, then a and b have same signy < 0, then a and b have different sign
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
ie.g. 1612 Find
Note: y > 0, then a and b have same signy < 0, then a and b have different sign
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
ie.g. 1612 Find 2 2
2 2
1220
a ba b
Note: y > 0, then a and b have same signy < 0, then a and b have different sign
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
ie.g. 1612 Find 2 2
2 2
1220
a ba b
2
2
2 8 4 2
aaa
Note: y > 0, then a and b have same signy < 0, then a and b have different sign
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
ie.g. 1612 Find 2 2
2 2
1220
a ba b
2
2
2 8 4 2
aaa
4b
Note: y > 0, then a and b have same signy < 0, then a and b have different sign
ORa ib x iy
2 2
2a b x
ab y
22 2 2 2 2 2
4 2 2 4 2 2
4 2 2 4
4
2 4 = 2
x y a b a b
a a b b a ba a b b
ibaiyx If2 2
2 2 2 2
then a b x
a b x y
ie.g. 1612 Find 2 2
2 2
1220
a ba b
2
2
2 8 4 2
aaa
4b
Note: y > 0, then a and b have same signy < 0, then a and b have different sign
ii 421612
( ) Find 5 4ii i
( ) Find 5 4ii i
2 2
2 2
5
41
a b
a b
( ) Find 5 4ii i
2 2
2 2
5
41
a b
a b
2
2
2 5 411 5 412
5 41 2
1 10 2 412
a
a
a
a
( ) Find 5 4ii i
2 2
2 2
5
41
a b
a b
2
2
2 5 411 5 412
5 41 2
1 10 2 412
a
a
a
a
2 1 5 41 521 5 412
5 41 2
1 10 2 412
b
b
b
( ) Find 5 4ii i
2 2
2 2
5
41
a b
a b
2
2
2 5 411 5 412
5 41 2
1 10 2 412
a
a
a
a
2 1 5 41 521 5 412
5 41 2
1 10 2 412
b
b
b
15 4 10 2 41 10 2 412
i i
2( ) Solve (2 ) (2 2 ) 0iii z i z i
2( ) Solve (2 ) (2 2 ) 0iii z i z i
22 2 4 2 22
2 4 4 1 8 82
2 5 122
i i iz
i i i
i i
2( ) Solve (2 ) (2 2 ) 0iii z i z i
22 2 4 2 22
2 4 4 1 8 82
2 5 122
i i iz
i i i
i i
5 12i 2 2
2 2
513
a ba b
2
2
2 8 4 2
aaa
3b
2( ) Solve (2 ) (2 2 ) 0iii z i z i
22 2 4 2 22
2 4 4 1 8 82
2 5 122
i i iz
i i i
i i
5 12i 2 2
2 2
513
a ba b
2
2
2 8 4 2
aaa
3b
2 2 32
i i
2( ) Solve (2 ) (2 2 ) 0iii z i z i
22 2 4 2 22
2 4 4 1 8 82
2 5 122
i i iz
i i i
i i
5 12i 2 2
2 2
513
a ba b
2
2
2 8 4 2
aaa
3b
2 2 32
i i
2 4 4 or 2 2 or 2 2
i i
i i
Cambridge: Exercise 1B; 1 to 4 ace etc, 5, 6, 7 cf,9, 10, 16 ac
