Can someone please tell me if my work and solution looks correct. I am a little unsure of whether or not I did it correctly.
Equation: sqrt(x+2) - x = 0
sqrt(x+2) = x
x + 2 = x^2
x^2 - x - 2 = 0
(x-2)(x+1) = 0
x = 2 and x = -1
-1 does not work, so the answer would be x = 2
-1 DOES work, if you take the (perfectly valid) negative square root of 1 in the original equation.
Both answers are correct
Thank you!
It looks ok to me.
2 works, but doesn't -1 work also?
sqrt(x+2)=x
sqrt(-1+2)=-1
sqrt(1)=-1
-1 or +1 = -1
by definition, the √ symbol means to take the positive square root of a number
so when you try to verify x = -1
LS = √(-1+2)-(-1)
= 1 + 1
= 2
which is not equal to RS
to see this, we could graph the corresponding function
f(x) = √(x+2) - x
this function is defined only for x≥-2, and has a single y-intercept of √2 and a single x-intercept of 2
so the only solution is x = 2
I learn something about math every time I answer one of these questions. I guess that's why I should stick to chemistry questions--but I can't answer all of them either.
<<by definition, the �ã symbol means to take the positive square root of a number>>
Well I'm not a math teacher, but I was never taught that
Nor was I in my algebra class of 1943. This COULD be a case of changing the rules (as has been done with all the SI units). A micron isn't a micron anymore (:(]. In fact my algebra teachers said, "DON'T forget there is a negative root of the square root of 4."
Not really changing the rules, more what is meant by terminology. My understanding is that the
principal square root function, Sqrt(x), always returns a positive value by definition (as mentioned above).
However, every positive number x has two square roots
one of which is +Sqrt(x)
and the other is -Sqrt(x).
Does this help?
To check if your work and solution are correct, let's go through the steps together.
The given equation is √(x+2) - x = 0.
Step 1: Isolate the square root term on one side:
Add x to both sides, which gives us:
√(x+2) = x.
Step 2: Square both sides of the equation to eliminate the square root:
(√(x+2))^2 = x^2.
(x+2) = x^2.
Step 3: Rearrange the equation so that it is in standard quadratic form:
x^2 - x - 2 = 0.
Step 4: Solve the quadratic equation either by factoring, completing the square, or using the quadratic formula.
In this case, you factored the quadratic equation as:
(x-2)(x+1) = 0.
Step 5: Set each factor equal to zero and solve for x:
x - 2 = 0, which gives x = 2.
x + 1 = 0, which gives x = -1.
Step 6: Confirm your solutions by substituting them back into the original equation:
For x = 2, √(2+2) - 2 = 0, which simplifies to √4 - 2 = 0, giving us 2 - 2 = 0, which is true.
For x = -1, √(-1+2) - (-1) = 0, which simplifies to √1 + 1 = 0, giving us 1 + 1 = 0, which is not true. Therefore, x = -1 is not a valid solution.
So, your solution stating x = 2 is correct. Well done!