Question 441834
First, square both sides:
{{{(sqrt(3x-5) - sqrt(x+2))^2 = 1^2}}}
Applying FOIL on the LHS we have:
{{{(3x-5) + (x+2) - 2sqrt((3x-5)(x+2)) = 1}}}
Collecting terms and isolating the term with the square root gives:
{{{4x-4 = 2sqrt((3x-5)(x+2))}}}
Isolating the square root term allows us to square both sides again:
{{{(4x-4)^2 = 4(3x-5)(x+2)}}}
Performing the multiplcation gives:
{{{16x^2 -32x + 16 = 4(3x^2 + x - 10)}}}
Divide through by 4 and collect terms:
{{{x^2 - 9x + 14 = 0}}}
Solve for x using the quadratic formula:
{{{x = (9 +- sqrt(81-56))/2}}} -> x = 2, 7
The squaring operation allows for positive and negative solutions, but only one value of x will make the original expression valid.
Check:
x=2: {{{sqrt(6-5) - sqrt(4) = 1 - 2 = -1}}} [Discard this solution]
x=7: {{{sqrt(21-5) - sqrt(9) = 4 - 3 = 1}}} [Correct]
Ans: x = 7