what will be it condition?
If IA B|=A² +B²
Answers
Answer:
There is a significant difference between principal square root and the square root. Principal square root is the unique, non-negative square root from a unique, non-negative real number.
The principal square root of 9 is 3. Not -3.
On the other hand, the square root is defined as Sqrt(n) = |n|. So the square root of 9 is +/- 3, or |3| (the absolute value of 3).
In this case, you are attempting to take the principal square root. This might be the way that you’re thinking of solving the problem:
(x+3)^2 = (x+7)^2
Sqrt((x+3)^2) = (x+3)
Sqrt((x+7)^2) =(x+7).
(x+3) cannot be (x+7), so no solution.
However, there is a solution to this problem. We know this because subtracting (x+3)^2 from both sides of the equation gives us:
(x+7)^2 - (x+3)^2 = 0
(x^2 + 14x + 49) - (x^2 + 6x + 9) = 0
8x + 40 = 0
8x = -40
x = -5
Checking this: (-5+7)^2 = 2^2 = 4 = (-2)^2 = (-5+3)^2
Why do we get a different answer this way? Because by multiplying, subtracting, and factoring we don’t lose a root, which is what occurs when you take the principal square root.
We actually want to avoid the principal square root when handling variable terms. You would reach the correct answer by taking the actual square root of both sides:
Sqrt((x+3)^2) = |x+3|
Sqrt((x+7)^2) = |x+7|
|x + 3| = |x + 7|
Experience with absolute value tells us that there are four possibilities: +/- (x+3) = +/- (x+7)
x+3 = x+7 → No solution
-(x+3) = -(x+7) → No solution (same as the first one if you multiply both sides by -1)
-(x+3) = x+7 → x = -5
x+3 = -(x+7) → x = -5 (same as the third one if you multiply both sides by -1)
Clearly, a solution exists. Notice anything about the approach(es) that yielded a satisfactory result? The sides to the equation had opposite signs. The principal square root omits the negative part of the square root. That is why you need to use the actual square root when handling variable terms.
Step-by-step explanation: