An operation * is defined on the set of real numbers by a * b = ab + 2(a + b + 1). find the identity elements
Answer Details
To find the identity elements of the operation *, we need to find values of x such that a * x = x * a = a for any value of a.
Let's start by considering a * x:
a * x = ax + 2(a + x + 1)
Now let's set this equal to a and solve for x:
ax + 2(a + x + 1) = a
ax + 2x + 2a + 2 = a
ax + 2x + a = -2
x(a + 2) = -2 - a
x = (-2 - a)/(a + 2)
So any value of x that satisfies this equation is an identity element of the operation *.
Now let's consider x * a:
x * a = xa + 2(x + a + 1)
We can substitute the value of x that we just found into this equation to get:
x * a = (-2 - a)a/(a + 2) + 2((-2 - a)/(a + 2) + a + 1)
Simplifying this expression, we get:
x * a = (-2a - a^2 - 4)/(a + 2)
So any value of x that satisfies both equations (a * x = a and x * a = a) is an identity element of the operation *.
To find the values of a that satisfy these equations, we can substitute x = (-2 - a)/(a + 2) into the equation a * x = a:
a * ((-2 - a)/(a + 2)) = a
-2 - a + 2 = a(a + 2)
a^2 + 3a + 2 = 0
(a + 1)(a + 2) = 0
So the values of a that satisfy the equation are -1 and -2.
Substituting these values into the equation for x that we found earlier, we get:
x = (-2 - (-1))/(-1 + 2) = -1
x = (-2 - (-2))/(-2 + 2) is undefined
Therefore, the identity element for this operation is -1.