convert into equation with 0 on right hand side
sq route (x+6) = 6-2 sq route (5-x)
To convert the given expression into an equation with 0 on the right-hand side, we need to manipulate the equation such that all terms are on the left-hand side.
Let's start by isolating the square root terms on one side of the equation:
√(x + 6) + 2√(5 - x) = 6
Next, we want to get rid of the square roots by squaring both sides of the equation:
(√(x + 6) + 2√(5 - x))^2 = 6^2
Expanding the left-hand side of the equation:
(x + 6) + 2(√(x + 6))(√(5 - x)) + 2(√(x + 6))(√(5 - x)) + 4(5 - x) = 36
Combining like terms:
x + 6 + 4(√(x + 6))(√(5 - x)) + 4(5 - x) = 36
Simplifying further:
x + 6 + 4(√(x + 6))(√(5 - x)) + 20 - 4x = 36
Now, we can combine the x terms and constants:
-3x + 26 + 4(√(x + 6))(√(5 - x)) = 36
To eliminate the square root terms, we need to square the equation once again:
(-3x + 26 + 4(√(x + 6))(√(5 - x)))^2 = 36^2
Expanding the left-hand side of the equation:
(-3x + 26)^2 + 2(-3x + 26)(4√(x + 6))(√(5 - x)) + (4√(x + 6))(√(5 - x))^2 = 1296
Combining like terms:
9x^2 - 156x + 676 + 2(-12x√(x + 6) + 104√(x + 6)√(5 - x)) + 4(x + 6)(5 - x) = 1296
Further simplification:
9x^2 - 156x + 676 - 24x√(x + 6) + 208√(x + 6)√(5 - x) + 20(x + 6) - 4x(x + 6) - 4x(5 - x) = 1296
Now, we combine like terms once again:
9x^2 - 156x + 676 - 24x√(x + 6) + 208√(x + 6)√(5 - x) + 20x + 120 - 4x^2 - 24x - 20x = 1296
Simplifying further:
5x^2 - 68x + 500 - 24x√(x + 6) + 208√(x + 6)√(5 - x) = 1296
Lastly, we move all terms to one side to have 0 on the right-hand side:
5x^2 - 68x + 500 - 24x√(x + 6) + 208√(x + 6)√(5 - x) - 1296 = 0
Therefore, the equation with 0 on the right-hand side is:
5x^2 - 68x - 24x√(x + 6) + 208√(x + 6)√(5 - x) - 796 = 0