help math essay

Question

When working with quadratics, we often use the identity a^(2)-b^(2)=(a+b)(a-b) to factor quadratic expressions of such a form. Using what you’ve learned about rationalizing denominators and complex conjugates, find and prove identities to easily factor expressions of these forms:

a. a^(2)+b^(2)
b. a^(2)-b
Hint: your factored forms may look very similar to the factored form of a^(2)-b^(2)

Answer 1

a. Two ways of doing (although rejected by purists)

 

1.

a² + b²

a² + 2ab + b² - 2ab            complete the square

(a + b)² - (√(2ab))²        

(a + b + √2ab)(a + b - √2ab)

 

2.

a² + b²

a² - (ib)²

(a + bi)(a - bi)

 

b.

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