Problem 3
I'm not sure what your teacher is asking here. Identities are equations, but your teacher didn't give you an equation.
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Problem 4
To show that we have an identity, I'm only going to manipulate the left hand side to transform it into the right hand side. The right hand side will remain the same the entire time.
(3x+7)^2 - 84x = (3x-7)^2
(3x+7)(3x+7) - 84x = (3x-7)^2
(9x^2+21x+21x+49) - 84x = (3x-7)^2
9x^2+42x+49-84x = (3x-7)^2
9x^2-42x+49 = (3x-7)^2
(3x-7)^2 = (3x-7)^2
The two formulas
- (a+b)^2 = a^2+2ab+b^2
- (a-b)^2 = a^2-2ab+b^2
may be handy in terms of expanding out (3x+7)^2 and factoring 9x^2-42x+49
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Problem 5
x + 1/x = 11
a + b = 11 .... where a = x and b = 1/x
(a+b)^2 = 11^2
a^2 + 2ab + b^2 = 121
x^2 + 2*x*(1/x) + (1/x)^2 = 121
x^2 + 2 + 1/(x^2) = 121
x^2 + 1/(x^2) = 121-2
x^2 + 1/(x^2) = 119
An alternative route is to solve x + 1/x = 11 for x. You'll get two solutions. Use either solution to plug into x^2 + 1/(x^2) to find the result is 119.
