I need help with this question. I need to know how to do the steps for it. (hope it makes sense)
Question:
Square root 2t+1 - 5 = - square root t
√(2t + 1) = 5 - √t
squaring ... 2t + 1 = t - 10√t + 25
subtracting ... t + 10√t - 24 = 0
let a = √t ... a^2 + 10a - 24 = 0
factoring ... (a + 12)(a - 2) = 0 ... a = -12 , a = 2 ... t = 144 , t = 4
substitute back to confirm solution(s)
I will read it as:
√(2t+1) - 5 = -√t
√(2t+1) = 5 - √t
square both sides:
2t+1 = 25 - 10√t + t
10√t = 24 - t
square again
100t = 576 - 48t + t^2
t^2 - 148t + 576 = 0
(t - 144)(t - 4) = 0
t= 144 or t = 4, BUT since we squared the equation each of
the answers must be verified in the original equation
if t = 144
LS = √(288+1) - 5 = 2
RS = -√144 = -12
≠ LS, so x = 144 does not work
if x = 5
LS = √9 - 5 = -2
RS = -2
so x = 4 is the only solution
To solve the equation **√(2t + 1) - 5 = -√(t)**, follow these steps:
Step 1: Isolate one of the square roots
First, isolate one of the square roots by moving the -√(t) term to the other side of the equation:
**√(2t + 1) = -√(t) + 5**
Step 2: Square both sides of the equation
Next, square both sides of the equation to eliminate the square roots:
**(√(2t + 1))^2 = (-√(t) + 5)^2**
Simplifying the equation gives:
**2t + 1 = t - 10√(t) + 25**
Step 3: Isolate the radical term
Move the terms containing the radical (√(t)) to one side of the equation and other terms to the other side:
**t + 10√(t) = 24**
Step 4: Isolate the radical term (continued)
To isolate the radical term, square both sides of the equation again:
**(t + 10√(t))^2 = 24^2**
Simplifying the equation gives:
**t^2 + 20t√(t) + 100t = 576**
Step 5: Isolate the radical term (continued)
Isolate the square root term by moving the other terms to the other side of the equation:
**20t√(t) = 576 - t^2 - 100t**
Step 6: Isolate the radical term (continued)
Divide both sides of the equation by 20t:
**√(t) = (576 - t^2 - 100t)/(20t)**
Simplify the right side of the equation:
**√(t) = (576 - t^2 - 100t)/20t**
Step 7: Solve for t
Square both sides of the equation again to eliminate the square root:
**t = ((576 - t^2 - 100t)/20t)^2**
Now, solve for t using algebraic manipulation, simplification, and factoring if possible.