(please help me to solve this question)
solving equation x+√(12p-x)/x-√(12p-x)= √(p)+1/√(p)-1 following roots are obtained
Answer:
The roots of the equation are x = 2√p and x = -2√p.
To solve this equation, we will use the following steps:
Step 1: Simplify the equation.
Step 2: Multiply both sides of the equation by the denominators to eliminate the square roots.
Step 3: Expand and simplify the equation.
Step 4: Move all the terms to one side of the equation.
Step 5: Factorize or solve for x.
Let's go through these steps in detail:
Step 1: Simplify the equation.
The given equation is:
x + √(12p - x)
_____________ = √(p) + 1
x - √(12p - x)
To simplify, let's convert the square roots into fractional exponents:
x + (12p - x)^(1/2)
__________________ = √(p) + 1
x - (12p - x)^(1/2)
Step 2: Multiply both sides of the equation by the denominators to eliminate the square roots.
(x - (12p - x)^(1/2))(x + (12p - x)^(1/2)) = (√p + 1)(x - (12p - x)^(1/2))
This step eliminates the square roots from both sides of the equation.
Step 3: Expand and simplify the equation.
Expanding both sides of the above equation, we have:
x^2 + (12p - x) - [(12p - x)^(1/2)]^2 = (x*√p - (12p - x)^(1/2)) + √p
Simplifying further, we get:
x^2 + 12p - x - (12p - x) = x√p - (12p - x)^(1/2) + √p
Step 4: Move all the terms to one side of the equation.
Rearranging the terms, we have:
x^2 + 12p - x - (12p - x) - (x√p - (12p - x)^(1/2) + √p) = 0
Expanding and simplifying this equation, we get:
x^2 + 12p - x - x√p + x^(1/2) - √p = 0
Step 5: Factorize or solve for x.
At this stage, it seems difficult to factorize or solve the equation algebraically due to the mixed terms involving x, x√p, and x^(1/2).
Hence, to find the roots of this equation numerically, we can use numerical methods such as the Newton-Raphson method or the bisection method. These methods involve iterating through the equation with initial guesses to converge to the roots.
Please note that since the equation involves the variable p, the roots obtained will be dependent on the value of p.
To solve the equation x + √(12p-x)/x - √(12p-x) = √(p) + 1/√(p) - 1, we will follow these steps:
Step 1: Simplify the equation by rationalizing the denominators in the left side of the equation.
First, let's deal with the denominator x - √(12p-x). We can multiply the numerator and denominator by the conjugate of the denominator to rationalize it.
(x - √(12p-x)) * (x + √(12p-x))
= x^2 - (12p-x)
= x^2 - 12p + x
Now the equation becomes:
x + √(12p-x) / (x^2 - 12p + x) = √(p) + 1/√(p) - 1
Step 2: Simplify the right side of the equation
√(p) + 1/√(p) - 1 can be written as a single fraction with a common denominator (√(p)):
= (√(p)*√(p) + 1 - √(p)) / √(p)
= (p + 1 - √(p)) / √(p)
Now the equation becomes:
x + √(12p-x) / (x^2 - 12p + x) = (p + 1 - √(p)) / √(p)
Step 3: Cross-multiply to get rid of the fractions.
(x + √(12p-x)) * √(p) = (p + 1 - √(p)) * (x^2 - 12p + x)
Step 4: Simplify both sides of the equation.
√(p)(x) + √(p)(√(12p-x)) = (p)(x^2) - (p)(12p) + (p)(x) + (x^2) - 12p + x - √(p)(x^2) + √(p)(12p) - √(p)(x)
Simplifying further, we get:
√(p)(√(12p-x)) = (p)(x^2) - (12p^2) + (x^2) - 12p + x - √(p)(x^2) + √(p)(12p)
Step 5: Combine like terms
√(p)(√(12p-x)) = -11p^2 + (2x^2) + (x) - 12p
Step 6: Square both sides of the equation to eliminate the radical (√) sign.
[√(p)(√(12p-x))]^2 = (-11p^2 + 2x^2 + x - 12p)^2
(p)(√(12p-x))^2 = (-11p^2 + 2x^2 + x - 12p)^2
Simplifying further, we get:
12p - x = (121p^4 - 22p^3x + 4x^4 - 132p^3 + 24p^2x - 4x^3 - 24px - 484p^2 + 88px - 16x^2 + 144p^2 + 26x - 36p)
Step 7: Rearrange the equation to obtain a quadratic equation.
0 = 121p^4 - 22p^3x + 4x^4 - 132p^3 + 24p^2x - 4x^3 - 12p^2 - 26x + 36p + 16x^2
This gives us a quadratic equation in terms of p and x. To solve for the values of p and x, we may need to use numerical methods or further simplifications depending on the specific values and restrictions of the variables in the equation.