if alfa +beta is equal to 5 alfabeta is equal to 2 then alfa-beta is equal ??
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Answer:
We are effectively told that
α
and
β
are the two roots of the quadratic equation:
x
2
=
5
x
−
3
Rearranged slightly, that is:
x
2
−
5
x
+
3
=
0
So we have:
0
=
x
2
−
5
x
+
3
0
=
(
x
−
α
)
(
x
−
β
)
0
=
x
2
−
(
α
+
β
)
x
+
α
β
Equating coefficients, that tells us that:
{
α
+
β
=
5
α
β
=
3
In addition, note that:
α
2
+
β
2
=
(
5
α
−
3
)
+
(
5
β
−
3
)
α
2
+
β
2
=
5
(
α
+
β
)
−
6
α
2
+
β
2
=
5
(
5
)
−
6
α
2
+
β
2
=
19
A quadratic equation with roots
α
β
and
β
α
is:
(
x
−
α
β
)
(
x
−
β
α
)
=
0
To clear the denominators and give us integer coefficients, we can multiply both sides by
α
β
to find:
0
=
α
β
(
x
−
α
β
)
(
x
−
β
α
)
0
=
(
β
x
−
α
)
(
α
x
−
β
)
0
=
α
β
x
2
−
(
α
2
+
β
2
)
x
+
α
β
0
=
3
x
2
−
19
x
+
3
So we can write a suitable quadratic equation as:
3
x
2
−
19
x
+
3
=
0
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