How do you solve #sqrt(3x+4)-sqrt(2x-7)=3#?

Question

Kennedy Adams · Accepted Answer

#x in {4, 64}#

We have:

#sqrt(3x +4) = 3 + sqrt(2x- 7)#

Squaring both sides, we get:

#(sqrt(3x+ 4))^2 = (3 + sqrt(2x - 7))^2#

#3x + 4 = 9 + 6sqrt(2x- 7) + 2x - 7#

Regrouping non-square root terms to one side of the equation, we get:

#x + 2 = 6sqrt(2x- 7)#

Square again:

#x^2 + 4x + 4 = 36(2x- 7)#

#x^2 +4x + 4 = 72x - 252#

#x^2 - 68x + 256 =0#

#(x -4)(x -64) = 0#

#x= 4 or 64#

Now test to see whether or not the solutions are valid.

Testing #x = 4#

#sqrt(3(4) + 4) =^? 3 + sqrt(2(4) - 7)#

#4 = 3 + 1 color(green)(√)#

Testing #x = 64#

#sqrt(3(64) + 4) =^? 3 + sqrt(2(64) - 7)#

#sqrt(196) =^? 3 + sqrt(121)#

#14 = 3 + 11 color(green)(√)#

So our solution set is #x in {4, 64}#.

Hopefully this helps!

Genesis Allen · Answer

undefined To solve the equation sqrt(3x+4)-sqrt(2x-7)=3, you can follow these steps:

1. Start by isolating one of the square root terms. Add sqrt(2x-7) to both sides of the equation:
   sqrt(3x+4) = sqrt(2x-7) + 3

2. Square both sides of the equation to eliminate the square root:
   (sqrt(3x+4))^2 = (sqrt(2x-7) + 3)^2

3. Simplify both sides of the equation:
   3x + 4 = (2x-7) + 6sqrt(2x-7) + 9

4. Combine like terms:
   3x + 4 = 2x + 2 + 6sqrt(2x-7) + 9

5. Simplify further:
   3x + 4 = 2x + 11 + 6sqrt(2x-7)

6. Move all terms involving x to one side and all constant terms to the other side:
   3x - 2x = 11 - 4 - 6sqrt(2x-7)

7. Simplify:
   x = 7 - 6sqrt(2x-7)

8. Square both sides of the equation again to eliminate the square root:
   x^2 = (7 - 6sqrt(2x-7))^2

9. Expand and simplify:
   x^2 = 49 - 84sqrt(2x-7) + 36(2x-7)

10. Simplify further:
   x^2 = 49 - 84sqrt(2x-7) + 72x - 252

11. Rearrange the equation:
   x^2 - 72x + 203 = -84sqrt(2x-7)

12. Square both sides of the equation one more time:
   (x^2 - 72x + 203)^2 = (-84sqrt(2x-7))^2

13. Expand and simplify:
   x^4 - 144x^3 + 412x^2 - 144x^3 + 20736x^2 - 59712x + 412x^2 - 59712x + 42849 = 7056(2x-7)

14. Simplify further:
   x^4 - 288x^3 + 62060x^2 - 119424x + 42849 = 14112x - 49428

15. Rearrange the equation:
   x^4 - 288x^3 + 62060x^2 - 14112x + 119424 - 49428 - 42849 = 0

16. Combine like terms:
   x^4 - 288x^3 + 62060x^2 - 14112x + 27147 = 0

17. At this point, the equation cannot be easily solved algebraically. You may need to use numerical methods or a graphing calculator to approximate the solutions.