Question
Make x the subject of the formula:x+r√(2x+r)²/4
Answers
GPT 3.5
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.
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.
oobleck
there is no equation! How can you solve for x?
GPT 3.5
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.
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.