A binary operation * is defined as the set of real numbers by a*b=a/b + b/a where a and b belong to R. Simplify (x+1)*2=3
According to the definition
(x+1)/2 + 2/(x+1) = 3
(x+1)^2 + 4 = 6(x+1)
x^2 + 2x + 1 + 4 = 6x+6
x^2 - 4x - 1 = 0
Since the question was to simplify , we could stop here, but
let's see what we get for x
this solves to get x = 2 ± √5
The bot never even used the fact that (x+1)/2 + 2/(x+1) was equal to 3
Apologies for the mistake in my previous response. You are correct – the simplification of the expression (x+1)*2=3 should lead to the quadratic equation x^2 - 4x - 1 = 0, as you have shown. And solving this quadratic equation does give the solutions x = 2 ± √5.
So the correct answer is indeed x = 2 ± √5, and my previous response was incorrect. Thanks for bringing this to my attention.
To simplify the expression (x + 1) * 2 = 3 using the given binary operation definition, we need to substitute the values of a and b and solve for x.
According to the definition, a * b = a/b + b/a.
Let's substitute a = (x + 1) and b = 2:
(x + 1) * 2 = (x + 1)/2 + 2/(x + 1)
Now, let's find a common denominator for the fractions:
(x + 1) * 2 = ((x + 1)(x + 1) + 4) / (2(x + 1))
Expanding the numerator:
(x + 1) * 2 = (x^2 + 2x + 1 + 4) / (2(x + 1))
Simplifying the numerator:
(x + 1) * 2 = (x^2 + 2x + 5) / (2(x + 1))
The equation is now:
(x^2 + 2x + 5) / (2(x + 1)) = 3
Next, we can cross-multiply:
(x^2 + 2x + 5) = 3 * 2(x + 1)
Simplifying further:
x^2 + 2x + 5 = 6(x + 1)
Expanding:
x^2 + 2x + 5 = 6x + 6
Rearranging terms:
x^2 - 4x - 1 = 0
At this point, we have a quadratic equation. We can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -4, and c = -1:
x = (-(-4) ± √((-4)^2 - 4(1)(-1))) / (2 * 1)
x = (4 ± √(16 + 4)) / 2
x = (4 ± √20) / 2
Simplifying the square root:
x = (4 ± √(4 * 5)) / 2
x = (4 ± 2√5) / 2
We can simplify further by canceling out the common factor of 2:
x = 2 ± √5
Therefore, the solutions to the equation (x + 1) * 2 = 3 are x = 2 + √5 and x = 2 - √5.
We start by substituting a = x + 1 and b = 2 into the definition of the operation:
(a * b) = a/b + b/a
(x + 1)*2 = (x + 1)/2 + 2/(x + 1)
Now we need to simplify this expression and solve for x. To do this, we will first get rid of the fractions in the expression by multiplying everything by the common denominator of 2(x + 1):
2(x + 1)(x + 1)*2 = 2(x + 1)(x + 1)/2 + 2*2(x + 1)/x + 1
Simplifying the left-hand side:
4(x + 1)^2 = 2(x + 1)^2 + 4(x + 1)
Expanding and simplifying the right-hand side:
2(x + 1)^2 = x^2 + 4x + 4
4(x + 1) = 4x + 4
Combining like terms on the right-hand side:
x^2 + 4x + 4 = 2(x + 1)^2 + 4(x + 1)
Expanding and simplifying the right-hand side:
x^2 + 4x + 4 = 2x^2 + 8x + 8
Subtracting x^2 + 4x + 4 from both sides:
0 = x^2 + 4x + 4 - 2x^2 - 8x - 8
Simplifying:
0 = -x^2 - 4x - 4
Multiplying both sides by -1:
0 = x^2 + 4x + 4
Factoring the left-hand side:
0 = (x + 2)^2
Taking the square root of both sides:
0 = x + 2
Subtracting 2 from both sides:
x = -2
Therefore, the solution to (x + 1)*2 = 3 is x = -2.