solve the equation
express the answers in p and q
x(square) - (p+q)x + (p+1)(q-1)=0
Then x=? or x=?
Recall that in a quadratic equation,
x^2 - (sum of roots)x + (product of roots) = 0
or if we let a & b the roots,
x^2 - (a+b) + (a)(b) = 0
From the given equation, we can see that the constant term is factored, which is (p+1)(q-1). Now, let's assume that the roots are p+1 and q-1. We then check its sum if it is equal to p+q. If it is, then they are the roots of the equation:
p + 1 + q - 1 = p + q
Therefore the roots are p+1 and q+1.
Another way to solve this is to use the quadratic equation:
x = [-b +/- sqrt(b^2 - 4ac)] / 2a
You should get the same answer.
Hope this helps~ :)
*I mean the roots are p+1 and q-1.
Just a typo error. Sorry about that. :3
To solve the equation x^2 - (p+q)x + (p+1)(q-1) = 0, we can use the quadratic formula. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's apply this formula to our equation and find the solutions:
In our equation, a = 1, b = -(p+q), and c = (p+1)(q-1).
x = (-(p+q) ± √((p+q)^2 - 4(1)(p+1)(q-1))) / (2(1))
Simplifying further:
x = (-(p+q) ± √(p^2 + 2pq + q^2 - 4(p+1)(q-1))) / 2
x = (-(p+q) ± √(p^2 + 2pq + q^2 - 4pq + 4p + 4 - 4q + 4)) / 2
x = (-(p+q) ± √(p^2 - 2pq + q^2 + 4p - 4q + 8)) / 2
Now, let's simplify the expression inside the square root:
x = (-(p+q) ± √((p-q)^2 + 4(p-1)(q+2))) / 2
Now we have the simplified expression for x in terms of p and q. The two solutions are:
x = (-(p+q) + √((p-q)^2 + 4(p-1)(q+2))) / 2
x = (-(p+q) - √((p-q)^2 + 4(p-1)(q+2))) / 2