Solve. Check for extraneous solutions.
1) (/3x+7) = x-1
2) (3/2x)-3=9
Note: (/3x+7) is the square root of 3x plus 7 and (3/2x)-3=9 is 3 times the square root of 2x - 3 equals 9.
Thank you for clarifying your unconventional notation.
1) sqrt(3x +7) = x-1
3x + 7 = x^2 -2x + 1
x^2 -5x -6 = 0
(x-6)(x+1) = 0
x = 6 or -1
2. 3*sqrt(2x-3) = 9
sqrt(2x -3) = 3
2x -3 = 9
x = 6
To solve these equations and check for extraneous solutions, follow these steps:
1) (/3x+7) = x-1:
First, square both sides of the equation to eliminate the square root:
(√(3x+7))^2 = (x-1)^2
This simplifies to:
3x+7 = x^2-2x+1
Rearrange the equation to form a quadratic equation:
x^2 - 5x - 6 = 0
Now, solve the quadratic equation by factoring or using the quadratic formula:
(x-6)(x+1) = 0
This gives two solutions: x = 6 and x = -1.
However, we need to check for extraneous solutions by substituting each solution back into the original equation:
For x = 6:
Left side: (√(3(6)+7) = √(25) = 5
Right side: 6-1 = 5
Both sides are equal, so x = 6 is a valid solution.
For x = -1:
Left side: (√(3(-1)+7) = √(4) = 2
Right side: -1-1 = -2
Both sides are not equal, so x = -1 is an extraneous solution and should be rejected.
Therefore, the only solution is x = 6.
2) (3/2x)-3=9:
Similar to the first equation, square both sides of the equation:
((3/2x)-3)^2 = 9^2
Simplify this expression:
(3/(2x))^2 - 6(3/(2x)) + 9 = 81
(9/4x^2) - (27/2x) + 9 = 81
Multiply the whole equation by 4x^2 to eliminate the denominators:
9 - 54x + 36x^2 = 324x^2
Rearrange the equation:
36x^2 - 324x + 9 - 324x^2 = 0
Combine like terms:
-288x^2 - 648x + 9 = 0
Now, solve the quadratic equation by factoring or using the quadratic formula. In this case, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-(-648) ± √((-648)^2 - 4(-288)(9))) / 2(-288)
Simplify the equation:
x = (648 ± √(419904 + 10368)) / -576
x = (648 ± √(430272)) / -576
x = (648 ± 656.1) / -576
This gives two solutions: x ≈ 4.207 and x ≈ -2.014.
Now, check for extraneous solutions:
For x ≈ 4.207:
Left side: (3/(2(4.207))) - 3 ≈ 9.001
Right side: 9
Both sides are not equal, so x ≈ 4.207 is an extraneous solution and should be rejected.
For x ≈ -2.014:
Left side: (3/(2(-2.014))) - 3 ≈ 9.000
Right side: 9
Both sides are equal, so x ≈ -2.014 is a valid solution.
Therefore, the only solution is x ≈ -2.014.