I have two questions here:
Solve the equation: ( (p + 1) / 2 ) ^ 2 - ( (p - 1) / 2 ) ^ 2
(Note it's P+1 over 2 for example but I think that's how you would write it on one line above?)
Factorize: x^2−x−2=0
((p+1)/2)^2 - ((p-1)/2)^2
This is just a^2-b^2 where a = (p+1)/2 and b = (p-1)/2
So that is (a+b)(a-b) = ((p+1)/2 + (p-1)/2)((p+1)/2)-(p-1)/2)
= (2p/2)(2/2) = p
x^2-x-2 = (x-2)(x+1)
Sure! Let's start with the first question:
To solve the equation: ( (p + 1) / 2 ) ^ 2 - ( (p - 1) / 2 ) ^ 2
Step 1: Simplify the equation
First, let's simplify the equation by expanding the expression using the difference of squares formula:
((p + 1) / 2)^2 - ((p - 1) / 2)^2 = (p^2 + 2p + 1) / 4 - (p^2 - 2p + 1) / 4
Next, we can combine the two fractions by finding a common denominator:
= (p^2 + 2p + 1 - p^2 + 2p - 1) / 4
After simplifying further, we get:
= (4p) / 4
= p
So, the solution to the equation ( (p + 1) / 2 ) ^ 2 - ( (p - 1) / 2 ) ^ 2 is p.
Now, let's move on to the second question:
To factorize the equation: x^2 - x - 2 = 0
Step 1: Split the middle term
We want to factorize the quadratic equation, so we need to rewrite the equation by breaking down the middle term (-x) into two terms whose coefficients multiply to give the last term (-2) and add or subtract to give the middle term (-x).
In this case, -2 can be factored as -2 * 1 or 2 * -1.
And -x can be written as -2x + x, which satisfies the conditions.
So, we rewrite the equation:
x^2 - x - 2 = x^2 - 2x + x - 2
Step 2: Group the terms
Now, let's group the terms:
(x^2 - 2x) + (x - 2)
Step 3: Factor by grouping
We can factor out the greatest common factor (GCF) from each group:
x(x - 2) + 1(x - 2)
Step 4: Combine the terms
Now, we have a common factor (x - 2) in both terms. We can combine these terms:
(x + 1)(x - 2) = 0
So, the factored form of the equation x^2 - x - 2 = 0 is (x + 1)(x - 2) = 0.
I hope this helps! Let me know if you have any further questions.