How to solve for "t" - step-by-step
sqrt(2t-1) + sqrt(3t+3) = 5
Thank you very much.
To solve for "t" in the equation sqrt(2t-1) + sqrt(3t+3) = 5, you can follow these step-by-step instructions:
Step 1: Isolate one of the square root terms
Begin by isolating one of the square root terms on one side of the equation. Let's isolate the term sqrt(2t-1) by subtracting sqrt(3t+3) from both sides:
sqrt(2t-1) = 5 - sqrt(3t+3)
Step 2: Square both sides of the equation
To eliminate the square root on the left side, square both sides of the equation. Remember that squaring both sides means multiplying each side by itself:
(sqrt(2t-1))^2 = (5 - sqrt(3t+3))^2
Simplifying this equation will give us:
2t-1 = (5 - sqrt(3t+3))^2
Step 3: Expand and simplify the right side
Expand the right side of the equation using the formula for a square of a binomial: (a - b)^2 = a^2 - 2ab + b^2.
2t-1 = 25 - 10sqrt(3t+3) + (3t+3)
Simplifying further will give:
2t-1 = 28 + 3t - 10sqrt(3t+3)
Step 4: Rearrange the equation
To simplify the equation further, rearrange it so that all terms involving "t" are on one side and all constant terms are on the other side:
2t - 3t = 28 + 10sqrt(3t+3) - 1
Simplifying this equation will give:
-t = 27 + 10sqrt(3t+3)
Step 5: Square both sides of the equation
To eliminate the square root term, square both sides of the equation:
(-t)^2 = (27 + 10sqrt(3t+3))^2
Simplifying this equation will give:
t^2 = 729 + 540sqrt(3t+3) + 100(3t+3)
Simplifying further will give:
t^2 = 729 + 540sqrt(3t+3) + 300t + 300
Step 6: Simplify the equation
Combine like terms in the equation:
t^2 - 300t = 729 + 540sqrt(3t+3) + 300
Rearrange the equation to isolate the square root term:
t^2 - 300t - 729 - 300 = 540sqrt(3t+3)
Simplifying further will give:
t^2 - 300t - 1029 = 540sqrt(3t+3)
Step 7: Square both sides of the equation
To eliminate the square root term, square both sides of the equation:
(t^2 - 300t - 1029)^2 = (540sqrt(3t+3))^2
Expanding both sides of the equation and simplifying will give a quadratic equation:
t^4 - 600t^3 + 459t^2 + 600t^3 - 360000t^2 + 582120t + 1029t^2 - 617940t - 1071540 = 291600(3t+3)
Simplifying further will give:
t^4 - 359571t^2 + 181200t - 1071540 = 875040t + 87480
Step 8: Simplify the equation
Combine like terms in the equation:
t^4 - 359571t^2 - 694840t - 1159020 = 875040t + 87480
Rearrange the equation to isolate all terms on one side:
t^4 - 360611t^2 - 1560880t - 1246500 = 0
Step 9: Solve the quadratic equation
Unfortunately, this equation cannot be easily solved using basic algebraic methods. You can apply numerical methods or use a graphing calculator to find the approximate solutions for "t".
These are the step-by-step instructions to solve for "t" in the equation sqrt(2t-1) + sqrt(3t+3) = 5.
√(3t+3) = 5 - √(2t-1), where clearly t ≥ 1/2
square both sides
3t+3 = 25 - 10√(2t-1) + 2t - 1
10√(2t-1) = 21 - t
square again
100(2t-1) = 441 - 42t + t^2
200t - 100 = 441 - 42t + t^2
t^2 - 242t + 541 = 0
t = (242 ± √56400)/2 = appr. 239.743 or 2.257
since we squared the equation all answers must be verified.
clearly x = 239.743 does not work
test x = 2.257
LS = 1.874 + 3.126 = 5
= RS
x = 2.257 or exactly 121 +10√141