I think I need help with this problem-
3+ square root(x-6) = square root of (x+9)
I thought I should square everything to get rid of square root signs
so I had 9 + x-6 = x+ 9
then I would have x+ 3once I subtracted 9-6 from the left side = x+9 on the right side which doesn't make any sense to move anything anymore it doesn't work what did I do wrong? Thanks
3 + √(x-6) = √(x+9)
you have to square the whole side, not just each term
9 + 6√(x-6) + x-6 = x+9
6√(x-6) = 6
√(x-6) = 1
square again
x-6 = 1
x = 7
check: This is MUST, since we squared each side
LS = 3 + √1 = 4
RS = √16 = 4
therefore x = 7
I'm probably just not understanding yet but Where did the 6 in front of square root of (x-6) come from and where did the 9 disappear to
left side squared
=(3+√(x-6) )^2
= (3+√(x-6))(3+√(x-6))
= 9 + 3√(x-6) + 3√(x-6) + x-6
= 9 + 6√(x-6) + x-6
right side squared = √(x+9) ^2 = x+9
so
9 + 6√(x-6) + x-6 = x+9 , collect all terms to right side, leave 6√.... on the left
6√(x-6) = x-x + 9-9 + 6
6√(x-6) = 6
divide both sides by 6
√(x-6) = 1
square again
x-6 = 1
etc
To solve the equation 3 + √(x - 6) = √(x + 9), you were correct in attempting to eliminate the square root signs by squaring both sides of the equation.
However, let's go through the steps again to identify where you made an error:
Original equation: 3 + √(x - 6) = √(x + 9)
1. Square both sides:
(3 + √(x - 6))^2 = (√(x + 9))^2
Expanding the left side:
(3 + √(x - 6)) * (3 + √(x - 6)) = x + 9
2. Apply the FOIL method to expand the left side:
9 + 3√(x - 6) + 3√(x - 6) + (x - 6) = x + 9
Combine like terms:
9 + 2√(x - 6) + (x - 6) = x + 9
3. Simplify the equation:
2√(x - 6) + x - 6 = x + 9
4. Now, isolate the term with the square root:
2√(x - 6) = 15
5. Divide the entire equation by 2:
√(x - 6) = 7.5
6. Square both sides again to eliminate the square root:
(√(x - 6))^2 = (7.5)^2
Simplifying:
x - 6 = 56.25
7. Add 6 to both sides:
x = 62.25
So, the solution to the equation is x = 62.25.
The mistake you made was when you squared the equation initially. When expanding (3 + √(x - 6))^2, you have to apply the FOIL method correctly.