√(5x-4) = 7 - √(13-x)
before we do anything, remember we cannot take the square root of a negative so,
5x-4≥0 and 13-x≥0
x≥4/5 and x ≤ 13
now back to the question,
square both sides
5x-4 = 49 - 14√(13-x) + 13-x
14√(13-x) = 66 - 6x
7√(13-x) = 33 - 3x
square it again:
49(13-x) = 1089 - 198x + 9x^2
9x^2-149x + 452 = 0
x = (149 ± √5929)/18
= (149 ± 77)/18
x = 95 or 4
We can rule out the x=95 since it falls outside our domain established above
So let's test x=4, (answers must be verified after a squaring process)
if x = 4
LS = √(20-4) = √16 = 4
RS = 7 - √9 = 7-3 = 4
x = 4
This looks like the reply I tried to send yesterday and wouldn't work.
My square roots signs, which didn't come our correctly are done on my Mac by using " Option V "
This has worked for me for years.
Here is the original reply I tried posting
√(5x-4) = 7 - √(13-x)
before we do anything, remember we cannot take the square root of a negative so,
5x-4≥0 and 13-x≥0
x≥4/5 and x ≤ 13
now back to the question,
square both sides
5x-4 = 49 - 14√(13-x) + 13-x
14√(13-x) = 66 - 6x
7√(13-x) = 33 - 3x
square it again:
49(13-x) = 1089 - 198x + 9x^2
9x^2-149x + 452 = 0
x = (149 ± √5929)/18
= (149 ± 77)/18
x = 95 or 4
We can rule out the x=95 since it falls outside our domain established above
So let's test x=4, (answers must be verified after a squaring process)
if x = 4
LS = √(20-4) = √16 = 4
RS = 7 - √9 = 7-3 = 4
x = 4
To solve the equation √(5x-4) = 7 - √(13-x), we start by noting that we cannot take the square root of a negative number. Therefore, we need to ensure that both 5x-4 and 13-x are greater than or equal to zero.
From 5x-4≥0, we can solve for x:
5x≥4
x≥4/5
From 13-x≥0, we can solve for x:
x≤13
So, we have x≥4/5 and x≤13 as our domain restrictions.
Now, let's proceed with solving the equation:
1. Square both sides of the equation:
(√(5x-4))^2 = (7 - √(13-x))^2
5x-4 = 49 - 14√(13-x) + (13-x)
2. Simplify the equation:
5x - 4 = 49 - 14√(13-x) + 13 - x
5x - 4 = 62 - 14√(13-x) - x
3. Rearrange the terms:
5x - x - 4 + 14√(13-x) = 62
4. Combine like terms:
4x + 14√(13-x) = 66
5. Divide both sides by 2:
2x + 7√(13-x) = 33
6. Move the term with the square root to one side:
7√(13-x) = 33 - 2x
7. Square both sides of the equation again:
(7√(13-x))^2 = (33 - 2x)^2
49(13-x) = 1089 - 132x + 4x^2
8. Simplify the equation:
637 - 49x = 1089 - 132x + 4x^2
9. Rearrange the terms:
4x^2 - 83x + 452 = 0
10. Solve for x by factoring or using the quadratic formula:
For this equation, the quadratic formula gives:
x = (83 ± √(83^2 - 4(4)(452))) / (2(4))
x = (83 ± √5929) / 8
x = (83 ± 77) / 8
11. Simplify the solutions:
x = 160/8 = 20
x = 6/8 = 3/4
12. Verify the solutions by substituting them back into the original equation:
For x = 20:
√(5(20)-4) = 7 - √(13-20)
√(100-4) = 7 - √(-7)
√96 = 7 - √(-7)
Since we cannot take the square root of a negative number, x = 20 is not a valid solution within our given domain restrictions.
For x = 3/4:
√(5(3/4)-4) = 7 - √(13-(3/4))
√(15/4 - 4) = 7 - √(13/4)
√(15/4 - (16/4)) = 7 - √(13/4)
√(-1/4) = 7 - √(13/4)
Again, since we cannot take the square root of a negative number, x = 3/4 is also not a valid solution within our given domain restrictions.
Therefore, the equation has no valid solutions within the given domain.