Solve.
(sqrt. x + 1) - (sqrt. 2x -12) = 1
The answers I reached were x = 8 and 24. If you plug in 8, you will get 1 as an answer. If you plug in 24, however, you will not get 1 as an answer.
Please see if 8 is the only solution. If not, please show me how to solve to find the other solutions.
After squaring twice, I also got x=8 or x=24
Whatever solutions we get after "squaring" both sides of an equation must be verified.
so for x = 8
Left Side = √(8+1) - √(16-12)
= 3 - 2
= 1
= right side
for x=24
left side = √24+1 - √(48-12)
= 5 - 6
= -1
which is NOT the right side.
so the only solution is x = 8
To solve the equation (sqrt. x + 1) - (sqrt. 2x - 12) = 1, we'll follow these steps:
Step 1: Isolate the square root terms on one side of the equation.
(sqrt. x + 1) = (sqrt. 2x - 12) + 1
Step 2: Square both sides of the equation to eliminate the square root terms.
(sqrt. x + 1)^2 = [(sqrt. 2x - 12) + 1]^2
Step 3: Simplify both sides of the equation.
x + 1 = 2x - 12 + 2(sqrt. 2x - 12) + 1
Step 4: Combine like terms and isolate the square root term.
x + 1 = 2x - 10 + 2(sqrt. 2x - 12)
Step 5: Move all terms to one side of the equation.
x - 2x = -10 - 1 + 2(sqrt. 2x - 12)
-x = -11 + 2(sqrt. 2x - 12)
Step 6: Simplify and isolate the square root term.
x = 11 - 2(sqrt. 2x - 12)
Step 7: Square both sides to eliminate the square root.
x^2 = (11 - 2(sqrt. 2x - 12))^2
Step 8: Simplify both sides.
x^2 = 121 - 44(sqrt. 2x - 12) + 4(2x - 12)
x^2 = 121 - 44(sqrt. 2x - 12) + 8x - 48
Step 9: Combine like terms.
x^2 - 8x + 44(sqrt. 2x - 12) = 73
At this point, the equation becomes a quadratic equation. We can solve it by setting it equal to zero and factoring or using the quadratic formula. However, the calculations become complex, and it is difficult to determine if the solutions will yield the desired result of x = 1.
Therefore, it is recommended to use a graphing calculator or software to visualize the equation and find the solutions. By graphing the equation y = x^2 - 8x + 44(sqrt. 2x - 12) - 73, you can determine the values of x that satisfy the equation.
Based on your initial solutions of x = 8 and 24, it seems that 8 is a valid solution, but 24 is not. To find other solutions, it is essential to use a graphing tool or calculator to evaluate any additional solutions to the quadratic equation.