solve x-sqrt(2x+1)=7
Can someone explain how to work this out. Im not exactly sure how you figure the square root part.
sqrt(2x+1) = (x-7)
square both sides
2x+1 = x^2 - 14 x + 49
x^2 - 16 x + 48 = 0
(x-4)(x-12) = 0
x = 4 or x = 12 MAYBE because when you square both sides you introduce an ambiguity since -a^2 = a^2 . Therefore check both answers in the original problem.
try x = 4
4 - sqrt (9) = 7
NO 4 -3 = 1, we need 4 + 3
try x = 12
12 - sqrt (25) = 12 -5 = 7 YES, we did it. x = 12 works.
x - 7 = sqrt(2x+1) ------>
(x-7)^2 = |2x+1| ------->
(x - 7)^2 = 2 x + 1 And 2x+1>=0 (1)
Or (x - 7)^2 = -2 x - 1 And 2x+1<0 (2)
(1):
x^2-16x+48 = 0 And 2x+1>=0 ------->
x = 4 or x = 12
(2): No solutions.
The solutions are thus x = 4 or x = 12
I forgot to check if x - 7 is positive. So, as Damon pointed out, x = 4 is not a solution.
To solve the equation x - √(2x + 1) = 7, we need to isolate the variable x on one side of the equation. Here's how you can work it out:
Step 1: Start with the given equation: x - √(2x + 1) = 7.
Step 2: To simplify the equation, let's first deal with the square root term (2x + 1). We need to isolate it.
Step 3: Add √(2x + 1) to both sides of the equation to get x = 7 + √(2x + 1).
Step 4: Now, we need to eliminate the square root term on the right side of the equation. To do this, we need to find a way to get rid of the square root.
Step 5: Square both sides of the equation (x = 7 + √(2x + 1)) to eliminate the square root. (x)^2 = (7 + √(2x + 1))^2.
Step 6: Simplify both sides of the equation. x^2 = 49 + 14√(2x + 1) + (2x + 1).
Step 7: Combine like terms on the right side of the equation. x^2 = 50 + 14√(2x + 1) + 2x.
Step 8: Now, let's isolate the square root term. Subtract 50 from both sides of the equation to get x^2 - 50 = 14√(2x + 1) + 2x.
Step 9: Move all the terms without the square root to the left side of the equation. x^2 - 50 - 2x = 14√(2x + 1).
Step 10: Now, square both sides of the equation again to eliminate the square root. (x^2 - 50 - 2x)^2 = (14√(2x + 1))^2.
Step 11: Simplify both sides of the equation. (x^2 - 50 - 2x)^2 = 196(2x + 1).
Step 12: Expand and simplify the left side of the equation. You can foil or use the formula (a - b)^2 = a^2 - 2ab + b^2.
Step 13: The equation should simplify to x^4 - 4x^3 + 104x^2 - 200x + 2500 = 392x + 196.
Step 14: Combine like terms on both sides. x^4 - 4x^3 + 104x^2 - 200x + 2500 - 392x - 196 = 0.
Step 15: Simplify further. x^4 - 4x^3 + 104x^2 - 592x + 2304 = 0.
Step 16: Unfortunately, this equation is a fourth-degree polynomial equation, which doesn't have a simple algebraic solution. You would need to use numerical methods, such as graphing or a computer program, to approximate the solutions.
Therefore, the equation x - √(2x + 1) = 7 does not have a simple algebraic solution.