If $$2a+3b = 12$$, $$2ab = 9$$, find the value of $$|2a-3b| $$

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Uniteasy Admin · Accepted Answer

$$2a+3b = 12$$n
Square both sides
$$(2a+3b)^2 = 100$$
Expand the LHS
$$4a^2+9b^2 +6\cdot 2ab = 100$$
then,
$$4a^2+9b^2 = 144-6\cdot 2ab = 144-6\cdot 9 =90$$
that is
$$4a^2+9b^2 = 90$$n
Construct an equation using the conjugate of $$2a+3b$$
$$(2a+3b)^2+(2a-3b)^2=2(4a^2+9b^2)$$
Substituting (1) and (2) results in
$$(2a-3b)^2 =2(4a^2+9b^2) - (2a+3b)^2$$
$$= 2\cdot 90-12^2 = 36 $$
Therefore
The value of $$|2a-3b|$$ is $$6$$

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