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Algebra/square root

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Question
If x^4 + lx^3+ mx^2+12x+9 is a perfect square
then find value of l and m? Here ^ denotes exponent.?
In exercise we have to find square root by division
or factorization method

Answer
Yes, the ^ symbol denotes an exponent.
To be a perfect square, we need (ax^2 + bx + c)^2 to be what's given.
Squaring this gives [1] ax^4 + 2abx^3 + (2ac+b)x^2 + 2bcx + c^2.

This says that since we have x^4, the value of a is 1.
This says that since we have a constant 9, and c^2 = 9, so c = 3 (or -3).

Putting in a=1 and c=3 into the [1] gives x^4 + 2bx^3 + (2+b)x^2 + 6bx + 9.
Since the value of the x term is 12x and it is also 2bx, this says 2b = 12, so b is 6.

Putting this in gives x^4 + 12x^3 + 8x^2 + 36x + 9.
This gives the value 12 to l and the value 8 to m.  

Algebra

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Scott A Wilson

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