Make x the subject of the formula:x+r√(2x+r)²/4
there is no equation! How can you solve for x?
Apologies for the confusion. Since there is no equation, but rather a formula, we can still manipulate it to isolate x.
Starting with the given formula:
x + r√(2x + r)²/4
Multiplying both sides of the equation by 4:
4x + r√(2x + r)² = 0
Now, we square both sides of the equation to eliminate the square root:
(4x + r√(2x + r)²)² = 0
Expanding the equation:
16x² + 8xr√(2x + r)² + r²(2x + r)² = 0
Simplifying further:
16x² + 8xr(2x + r) + r²(2x + r)² = 0
Now, we can simplify and rewrite the equation:
16x² + 16x²r + 8xr² + r²(2x + r)² = 0
Combine like terms:
32x² + 8xr² + r²(2x + r)² = 0
Unfortunately, it is not possible to explicitly solve for x in this resulting equation. The equation is now nonlinear and involves multiple terms. Thus, we cannot make x the subject of the formula in a simple manner.
To make x the subject of the formula, we will go through the following steps:
1. Distribute the exponent to the expression inside the square root:
x + r √(2x + r)² / 4 = x + r (2x + r) / 4
2. Multiply both sides of the equation by 4 to remove the denominator:
4(x + r √(2x + r)² / 4) = 4(x + r (2x + r) / 4)
4x + 4r √(2x + r)² = x + r (2x + r)
3. Move all terms containing x to the left side of the equation and the remaining terms to the right side:
4x - x = -4r √(2x + r)² - r (2x + r)
3x = -4r √(2x + r)² - r (2x + r)
4. Square both sides of the equation to eliminate the square root:
(3x)² = (-4r √(2x + r)² - r (2x + r))²
9x² = 16r²(2x + r)² + 8r³(2x + r) + r²(2x + r)²
5. Expand and simplify the equation:
9x² = 16r²(4x² + 4rx + r²) + 8r³(2x + r) + r²(4x² + 4rx + r²)
9x² = 64r²x² + 64r³x + 16r⁴ + 16r³x + 8r⁴ + 4r²x² + 4r³x + r⁴
6. Combine like terms:
9x² = 68r²x² + 80r³x + 20r⁴
7. Move all terms to one side of the equation, leaving zero on the other:
68r²x² + 80r³x + 20r⁴ - 9x² = 0
8. Now we have a quadratic equation in terms of 'x'. To solve for 'x', we can factor or use the quadratic formula. Solving this equation will give us the value(s) of 'x'.
To make x the subject of the formula, we will need to isolate x on one side of the equation.
Starting with the given formula:
x + r√(2x + r)²/4
To get rid of the square root, we need to square both sides of the equation:
(x + r√(2x + r)²/4)² = x²
Expanding the square on the left side of the equation:
x² + 2xr√(2x + r) + (r√(2x + r)²/4)² = x²
Simplifying the right side of the equation:
x² + 2xr√(2x + r) + (r²(2x + r)/4) = x²
Subtracting x² from both sides of the equation:
2xr√(2x + r) + (r²(2x + r)/4) = 0
Moving all terms involving x to one side of the equation:
2xr√(2x + r) = -(r²(2x + r)/4)
Dividing both sides of the equation by 2r:
√(2x + r) = -(r(2x + r)/8r)
Squaring both sides of the equation to eliminate the square root:
(2x + r) = (r²(2x + r)²/64r²)
Expanding the equation:
2x + r = (4x² + 4xr + r²)/64r²
Multiplying both sides of the equation by 64r²:
128xr² + 64r³ = 4x² + 4xr + r²
Re-arranging the equation:
0 = 4x² + (-128xr² - 4xr + 64r³) + r²
Simplifying further:
0 = 4x² - 132xr² + 64r³ + r²
At this point, it is not possible to directly solve for x as the equation is a polynomial of degree 2 with additional terms. Therefore, it is not possible to make x the subject of the formula in a simple manner.