square root (x-3)-square root(x)=3
can someone please solve this for me
?(x-3) - ?x = 3
?(x-3) = ?x + 3
square both sides:
x-3 = x + 6?x + 9
-12 = 6?x
?x = -2
by definition of ? , this has no solution.
proceeding anyway ...
square both sides again:
x = 4
since we squared, all answers must be verified:
if x = 4,
LS = ?1 - ?4
= 1 - 2
= -1
? RS
There is no solution.
verification:
http://www.wolframalpha.com/input/?i=solve+%E2%88%9A(x-3)+-+%E2%88%9Ax+%3D+3
Look at it on a number line.
for positive x-3
x must be to the right of 3
but
then sqrt x must be greater than sqrt (x-3)
which is not possible.
If it were -3 on the right, that would be different
(x-3)^.5 - x^.5 = 3
square both sides
[(x-3)^.5-x^.5][(x-3)^.5-x^.5] = 9
(x-3)-2(x^.5(x-3)^.5)+x = 9
2x -2(x^.5(x-3)^.5)=12
(x^2-3x)^.5 = 6-x
x^2-3x = 36 -12x + x^2
9 x = 36
x = 4
but obviously
1 - 2 is not 3
Sure! Let's solve the equation step by step.
First, let's start by isolating one of the square roots. We'll choose the square root (x-3). Move the other square root, -sqrt(x), to the other side of the equation:
sqrt(x-3) = sqrt(x) + 3
Next, square both sides of the equation to eliminate the square roots:
(x-3) = (sqrt(x) + 3)^2
Now, let's simplify the right side of the equation by squaring the binomial (sqrt(x) + 3):
(x-3) = (sqrt(x))^2 + 2(sqrt(x))(3) + 3^2
(x-3) = x + 6(sqrt(x)) + 9
Now, combine like terms on the right side of the equation:
x - 3 = x + 6(sqrt(x)) + 9
Move all terms involving x to one side of the equation:
x - x - 3 = 6(sqrt(x)) + 9
Simplify by subtracting x from both sides:
-3 = 6(sqrt(x)) + 9
Next, isolate the expression involving the square root:
6(sqrt(x)) = -3 - 9
Simplify by combining the constants on the right side:
6(sqrt(x)) = -12
Divide both sides of the equation by 6:
sqrt(x) = -2
At this point, we can see that there is a contradiction since the square root function always returns non-negative values. Therefore, there is no solution to the equation.
Hence, the equation sqrt(x-3) - sqrt(x) = 3 has no solution.