slove the equestionx+√x=6/25
sqrt(x)=6/25
x = ( 6 / 25 ) ^ 2
x = 36 / 625
x+sqrt(x)=6/25
sqrt(x) = 6/25-x
x=(6/25)^2 - 12x/25+x^2
rearrange to a quadratic
0=x^2-37x/25+36/625
now solve for x
To solve the equation x + √x = 6/25, we can follow these steps:
Step 1: Start by isolating the square root term. Subtract √x from both sides of the equation:
x - √x + √x = 6/25 - √x
Simplifying, we get:
x = 6/25 - √x
Step 2: Now, let's isolate the square root term on the right side of the equation. Since we cannot directly combine a square root term with a non-square root term, we need to eliminate the square root. To do this, we'll square both sides of the equation:
(x)^2 = (6/25 - √x)^2
Simplifying further, we get:
x^2 = (6/25)^2 - 2(6/25)(√x) + (√x)^2
Step 3: Simplify the expression on the right side of the equation. Recall that (√x)^2 = x:
x^2 = 36/625 - 2(6/25)(√x) + x
Step 4: Rearrange the equation so that all terms involving x are on one side, and all constants are on the other side:
x^2 - x + 2(6/25)(√x) - 36/625 = 0
Step 5: Now comes the tricky part. The equation is quadratic in nature, but it also involves a square root. To simplify it, we can substitute another variable. Let's set √x = t. This means x = t^2.
Substituting these values into the equation, we get:
(t^2)^2 - t^2 + 2(6/25)(t) - 36/625 = 0
Step 6: Simplify the equation:
t^4 - t^2 + (12/25)t - 36/625 = 0
Step 7: At this point, we can try to factor or use the quadratic formula to solve for t. However, factoring or using the quadratic formula may not be easy due to the complex terms involved. In such cases, numerical methods, like using a graphing calculator or numerical approximation techniques, can be used to find the solution.
For example, you can graph the function y = t^4 - t^2 + (12/25)t - 36/625 and find the x-intercepts (where the graph crosses the x-axis). These x-intercepts will correspond to the values of t that satisfy the equation.
Alternatively, you can use a numerical approximation method, such as Newton's method or the bisection method, to approximate the roots of the equation. These methods involve iterative calculations to refine the solution until a desired level of accuracy is achieved.
So, to find the exact solution for x in the equation x + √x = 6/25, you may need to use numerical methods or approximations.