√3x-4=2√x-5
Solve this equation, if there is more then one answer separate with a comma
To solve the equation √(3x) - 4 = 2√x - 5, follow these steps:
Step 1: Isolate the square root terms on one side of the equation.
Add 4 to both sides:
√(3x) = 2√x - 1
Step 2: Square both sides of the equation to eliminate the square root.
(√(3x))^2 = (2√x - 1)^2
3x = 4x - 4√x + 1
Step 3: Move all terms to one side of the equation to find a quadratic equation.
Move all terms to the left side:
3x - 4x + 4√x - 1 = 0
-x + 4√x - 1 = 0
Step 4: Solve for √x by treating it as a single variable.
Let u = √x, then the equation becomes:
-u^2 + 4u - 1 = 0
Step 5: Solve the quadratic equation.
You can either factor the quadratic equation or use the quadratic formula. Factoring may not always be possible, so we'll solve it using the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -1, b = 4, and c = -1.
u = (-(4) ± √((4)^2 - 4(-1)(-1))) / 2(-1)
Simplify:
u = (-4 ± √(16 - 4)) / -2
u = (-4 ± √(12)) / -2
u = (-4 ± 2√3) / -2
Simplify further:
u = (2(√3) - 4) / -2 OR u = (-2(√3) - 4) / -2
Step 6: Convert back to x.
Since u = √x, we need to square the values we found for u to get the values for x.
For u = (2√3 - 4) / -2:
√x = (2√3 - 4) / -2
Squaring both sides:
x = ((2√3 - 4) / -2)^2
For u = (-2√3 - 4) / -2:
√x = (-2√3 - 4) / -2
Squaring both sides:
x = ((-2√3 - 4) / -2)^2
Simplify and compute to get the final solutions for x.