solving equation 7√(x/1-x) + 8√(1-x/x) = 15 following roots are obtained
64/113,1/2
so, you gonna share those roots with us?
To solve the equation 7√(x/1-x) + 8√(1-x/x) = 15, you need to manipulate the equation to isolate the variable x. Here's the step-by-step process:
1. Start by isolating one of the square roots. Let's start with the first square root term.
7√(x/1-x) = 15 - 8√(1-x/x)
2. Square both sides of the equation to get rid of the square root.
(7√(x/1-x))^2 = (15 - 8√(1-x/x))^2
3. Simplify the left side by squaring both the coefficient and the square root.
49 * (x/1-x) = (15 - 8√(1-x/x))^2
4. Expand the right side using the formula (a - b)^2 = a^2 - 2ab + b^2.
49 * (x/1-x) = 225 - 240√(1-x/x) + 64 * (1-x/x)
5. Simplify the right side by multiplying and simplifying the rational expression.
49 * (x/1-x) = 225 - 240√(1-x/x) + 64 * (1-1/x)
49 * (x/1-x) = 225 - 240√(1-x/x) + 64 * (0)
49 * (x/1-x) = 225 - 240√(1-x/x)
6. Multiply both sides of the equation by (1-x) to eliminate the denominator.
49x = 225(1-x) - 240√(1-x/x)(1-x)
7. Simplify the right side by distributing and collecting like terms.
49x = 225 - 225x - 240(1-x)√(1-x/x)
49x = 225 - 225x - 240√(1-x)
8. Move all terms involving x to one side of the equation and the remaining terms to the other side.
49x + 225x = 225 - 240√(1-x)
(49 + 225)x = 225 - 240√(1-x)
274x = 225 - 240√(1-x)
9. Divide both sides by 274 to solve for x.
x = (225 - 240√(1-x)) / 274
This equation represents the solution to the given equation. Since it involves a square root term, it may not have an exact solution that can be expressed in a simple form. To obtain the numerical roots, you can use numerical methods such as substitution, iteration, or a numerical solver.