I need help with this algebra problem....T___T?

Question

Factor each expression as a difference of two squares:
a^2 - (x+1)^2

What's 'a difference of two squares', and how would you use it in this problem?

How about the following question?

(c-d)^2 - e^2

I'm really confused right now, so please help!!!!!!!!
Thanks...
>.<

I'm not looking to find ROOTS or SOLUTIONS, but just how to factor it...
Thanks for the help!!! :]

Answer 1

A difference of squares is when A^2 - B^2 = (A-B)(A+B)

So, A^2 - (X+1)^2 = (A - (X+1))(A+(X+1))

Think of it as X+1 = B.

Similarly, for the second one, it would be:
(C-D)^2 - E^2 = ((C-D) - E)((C-D) + E)

Here you would think of (C-D) = B.

Hope this helps!

Answer 2

Let's say you have a quadratic equation like

x^2 + 6x + 5

Take the second term and divide it by 2

6 / 2 = 3

Take that number and square it.

3^2 = 9

Go back to the original equation and separate the x terms from non-x terms.

(x^2 + 6x) + 5

Now add 9 inside the parentheses and subtract it outside the parentheses.

(x^2 + 6x + 9) + 5 - 9

(x^2 + 6x + 9) - 4

The first part of the equation is (x + 3)^2. The second part subtracts 2^2.

So write the equation as:

(x + 3)^2 - 2^2 = 0

Later you will learn to move the 2^2 to the other side of the equals sign and get

(x + 3)^2 = 2^2

When we take the root of either side, we get

x + 3 = +/- 2

x = +/- 2 - 3

x = -1 or -5

If you were to use the quadratic equation, you would get the same answers.

Answer 3

a^2 - (x+1)^2
This is a squared minus (x plus 1) squared. As both quantities are something squared, that makes the expression a difference of two squared numbers, or the diffference of two squares.
A difference of two squares a^2 - b^2 can be factorised into (a + b)(a - b).
Multiply those brackets together, and you get:
a^2 + ab - ab - b^2
The ab terms disappear, as one is minus the other.
That leaves a^2 - b^2 ........(1)

Using this principle, you can factorise a^2 - (x + 1)^2 as:
( a^2 + (x + 1) )( a^2 - (x + 1) )
That is treating x + 1 the same as b in (1) above.
Now simplify the factors by eliminating the inner brackets. That gives:
(a^2 + x + 1)(a^2 - x - 1).


Using the same principle, treating c - d as a in (1) above:
(c-d)^2 - e^2
= ( (c - d) + e )( (c - d) - e )
Eliminating the inner brackets leaves:
(c - d + e)(c - d - e).

Answer 4

The term is "the sum AND the difference of two squares". If you have a quadratic in the form
(cx)^2 - d^2, where c and d are positive and integral, the quadratic factors to (cx+d)(cx-d). If you multiply these together (FOIL), the (OI) terms cancel each other out.

Answer 5

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Answer 6

with the first, presumably this is really a^2-(x+1)^2=0?
then a^2=(x+1)^2
taking square roots
a= +or- (x+1)
similarly for the second

Answer 7

A^2 - B^2 = (A-B)*(A+B)

have fun ;)