(square root of x)+(square root of x-5) = 1
To solve the equation (square root of x) + (square root of x-5) = 1, we can follow these steps:
Step 1: Identify the radicals
Notice that we have two square roots in the equation: the square root of x and the square root of x-5.
Step 2: Isolate one square root
To simplify the equation, let's isolate one of the square roots. Since we have two square roots, it doesn't matter which one we choose to isolate. Let's isolate the square root of x.
Start by subtracting (square root of x-5) from both sides of the equation:
(square root of x) = 1 - (square root of x-5)
Step 3: Square both sides
To eliminate the square root, we need to square both sides of the equation. Squaring both sides will allow us to get rid of the radical sign on the left side.
(square root of x)^2 = (1 - (square root of x-5))^2
Simplifying the left side gives us:
x = (1 - (square root of x-5))^2
Step 4: Expand and simplify
On the right side, we have a square of a binomial. We can expand it using the formula (a-b)^2 = a^2 - 2ab + b^2.
x = 1^2 - 2(1)((square root of x-5)) + ((square root of x-5))^2
x = 1 - 2(square root of x-5) + x-5
Simplifying further, we get:
x - x = -2(square root of x-5) - 4
-2 = -2(square root of x-5)
Step 5: Solve for x
Divide both sides of the equation by -2 to isolate the square root:
(square root of x-5) = -2 / -2
(square root of x-5) = 1
Now square both sides to eliminate the square root:
(x-5) = 1^2
x-5 = 1
x = 1 + 5
x = 6
So the solution to the equation is x = 6. Remember to always check your answer by substituting it back into the original equation to ensure it satisfies the equation.