Solve the following equation
1) root(2x+4)= root(6x+1) -1
Square both sides to get:
2x+4 = 6x+1 - 2√(6x+1) + 1
transpose terms without the square-root radical to the left, and everything else to the right and divide both sides by the common factor 2:
2√(6x+1) = 6x+1 -(2x+4) + 1
√(6x+1) = 2x - 1
Square again:
6x+1 = 4x² - 4x + 1
4x² -10x=0
x(x-5)=0
Solve for x to get x=0 or x=5
Since we squared two times, it is possible that we have introduced solutions which are not acceptable. To eliminate the invalid solutions, substitute each value into the original equation and check if the solution works.
x=0 gives √(4)=√(1)-1
which clearly does not work, so it is rejected.
x=5/2 gives √(5+4)=√(15+1)-1
which is valid.
So x=5/2.
To solve the equation root(2x + 4) = root(6x + 1) - 1, we can follow these steps:
Step 1: Remove the square roots by squaring both sides of the equation.
(root(2x + 4))^2 = (root(6x + 1) - 1)^2
Step 2: Simplify the equation.
2x + 4 = (6x + 1) - 2(root(6x + 1)) + 1
Step 3: Combine like terms.
2x + 4 = 6x - 2(root(6x + 1))
Step 4: Move all terms containing x to one side of the equation.
2x - 6x = -2(root(6x + 1)) - 4
Step 5: Simplify the equation.
-4x = -2(root(6x + 1)) - 4
Step 6: Divide the entire equation by -4 to solve for x.
x = (-2(root(6x + 1)) - 4) / -4
Step 7: Simplify the equation further if needed.
Please note that this solution is derived by following a series of algebraic manipulations.