3)Make x the subject of the formula x+r=√[(2x+r)^2/4]
To make x the subject of the given formula, we need to isolate x on one side of the equation.
First, let's square both sides of the equation:
(x + r)^2 = [(2x + r)^2]/4
Expanding the square on the left side:
x^2 + 2xr + r^2 = [(2x + r)^2]/4
Multiplying both sides by 4 to eliminate the fraction:
4(x^2 + 2xr + r^2) = (2x + r)^2
Expanding the square on the right side:
4(x^2 + 2xr + r^2) = (4x^2 + 4xr + r^2)
Distributing 4 on the left side:
4x^2 + 8xr + 4r^2 = 4x^2 + 4xr + r^2
Subtracting 4x^2 from both sides:
8xr + 4r^2 = 0 + 4xr + r^2
Subtracting 4xr and r^2 from both sides:
8xr - 4xr + 4r^2 - r^2 = 0
Simplifying:
4xr + 3r^2 = 0
Factor out an r from the left side:
r(4x + 3r) = 0
Now we have two possible solutions:
1) r = 0
If r = 0, we can substitute it back into the original equation:
x + r = √[(2x + r)^2/4]
x + 0 = √[(2x + 0)^2/4]
x = √[4x^2/4]
x = √x^2
x = |x| (Taking the positive square root since x cannot be negative)
So one solution is x = 0.
2) 4x + 3r = 0
If 4x + 3r = 0, we can substitute it back into the original equation:
x + r = √[(2x + r)^2/4]
x + (-4x/3) = √[(2x + (-4x/3))^2/4]
x - (4/3)x = √[(2x - (4x/3))^2/4]
(3/3)x - (4/3)x = √[(2x - (4x/3))^2/4]
(3x - 4x)/3 = √[(2x - 4x/3)^2/4]
(-x)/3 = √[(2x - 4x/3)^2/4]
(-x)/3 = √[(6x/3 - 4x/3)^2/4]
(-x)/3 = √[(2x/3)^2/4]
(-x)/3 = √[4x^2/9]/2
(-x)/3 = √(4x^2/36)
(-x)/3 = (2/6)x
(-x)/3 = (1/3)x
Multiply both sides by 3 to eliminate the denominator:
-x = x
Subtract x from both sides:
-x - x = 0
-2x = 0
Divide by -2 (or multiply by -1/2) to solve for x:
(-1/2)(-2x) = (-1/2)(0)
x = 0
So the other solution is x = 0.
In conclusion, the given formula x + r = √[(2x + r)^2/4] has one solution x = 0 when r = 0, or x = 0 when 4x + 3r = 0.
To make x the subject of the formula x + r = √[(2x + r)^2/4], we can follow these steps:
Step 1: Square both sides of the equation to eliminate the square root:
(x + r)^2 = [(2x + r)^2/4]
Step 2: Expand the squared terms:
(x + r)(x + r) = [(2x + r)(2x + r)]/4
Step 3: Multiply the brackets using the distributive property:
x^2 + 2xr + r^2 = (4x^2 + 4xr + r^2) / 4
Step 4: Simplify the right side of the equation:
x^2 + 2xr + r^2 = 4x^2/4 + 4xr/4 + r^2/4
Step 5: Combine like terms:
x^2 + 2xr + r^2 = x^2 + xr + r^2/4
Step 6: Cancel out the common terms on both sides of the equation:
2xr = xr + r^2/4
Step 7: Subtract xr from both sides:
2xr - xr = r^2/4
Step 8: Simplify the left side:
xr = r^2/4
Step 9: Divide both sides of the equation by r:
x = r^2/ (4r)
Therefore, x = r/4.