Solve 3x^1/2 + 5 - 2x^-1/2 = 0
3√x + 5 - 2/√x = 0
let y = √x
3y + 5 - 2/y = 0
multiply each term by y
3y^2 + 5y - 2 = 0
(3y - 1)(y + 2) = 0
y = 1/3 or y = -2
then √x = 1/3 , x = 1/9
or
√x = -2 , x is not a real number
x = 1/9 is the only real solution.
Slove the following equation 7(2x+1)-3-2x
Ah, it seems like you've got an algebraic equation on your hands! But fear not, I'm here to help you solve it, with a little touch of humor, of course.
Let's break down the equation:
3x^(1/2) + 5 - 2x^(-1/2) = 0
To make things simpler, let's denote x^(1/2) as a ghost (boo!) and x^(-1/2) as a zombie (groan!).
Now, let's re-write the equation in terms of our friendly "ghost" and "zombie":
3(ghost) + 5 - 2(zombie) = 0
We want to isolate our ghosts and zombies from the rest of the equation, so let's move the numbers around first:
3(ghost) - 2(zombie) = -5
Next, let's separate the ghosts and zombies:
3(ghost) - 2(zombie) = -5
Now, we can bring a bit of humor into this equation. Since we're dealing with ghosts and zombies, it seems only fitting that we make them equal to each other, right?
So, let's say ghost = zombie. Now our equation becomes:
3(ghost) - 2(ghost) = -5
Alright, let's simplify:
ghost = -5
But remember, ghost is actually x^(1/2), so let's replace it:
x^(1/2) = -5
Now, to get rid of the ghost (no ghostbusting required), we need to square both sides:
(x^(1/2))^2 = (-5)^2
This simplifies to:
x = 25
So, my friend, the solution to the equation 3x^(1/2) + 5 - 2x^(-1/2) = 0 is x = 25. I hope this quirky journey through ghosts and zombies made the process a bit more enjoyable for you!
To solve the equation 3x^(1/2) + 5 - 2x^(-1/2) = 0, we can follow these steps:
Step 1: Let's begin by moving the constant term (5) to the other side of the equation.
3x^(1/2) - 2x^(-1/2) = -5
Step 2: Next, let's multiply through by x^(1/2) to eliminate the fractional exponents.
3x^(1/2) * x^(1/2) - 2x^(-1/2) * x^(1/2) = -5 * x^(1/2)
Simplifying this equation gives us:
3x - 2 = -5 * x^(1/2)
Step 3: Now, let's isolate the terms with x^(1/2) on one side by moving the constant term (-2) to the other side of the equation.
3x = -5 * x^(1/2) + 2
Step 4: Next, let's square both sides of the equation to eliminate the remaining fractional exponent.
(3x)^2 = (-5 * x^(1/2) + 2)^2
Simplifying using the FOIL method gives us:
9x^2 = (25 * x) + (20 * x^(1/2)) + 4
Step 5: Let's rearrange the terms to bring all terms to one side of the equation.
9x^2 - 25x - 20x^(1/2) - 4 = 0
Step 6: Now we can solve this quadratic equation.
Unfortunately, this equation cannot be easily factored. To find the solution, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 9, b = -25, and c = -20.
Plugging these values into the quadratic formula, we get:
x = (-(-25) ± √((-25)^2 - 4 * 9 * (-20))) / (2 * 9)
Let's simplify this:
x = (25 ± √(625 + 720)) / 18
Simplifying further, we have:
x = (25 ± √1345) / 18
Therefore, the solutions to the equation 3x^(1/2) + 5 - 2x^(-1/2) = 0 are:
x = (25 + √1345) / 18
x = (25 - √1345) / 18
To solve the equation 3x^(1/2) + 5 - 2x^(-1/2) = 0, we can follow these steps:
Step 1: Identify any terms with the same base.
In this equation, we have terms with the base x^(1/2) and x^(-1/2).
Step 2: Rewrite the equation using a common base.
Since x^(1/2) is the same as the square root of x, and x^(-1/2) is the reciprocal of the square root of x, we can rewrite the equation as:
3√x + 5 - 2 / √x = 0
Step 3: Combine like terms.
We can simplify the equation further by combining the terms involving the square root of x. The equation now becomes:
3√x - 2 / √x + 5 = 0
Step 4: Multiply the entire equation by the least common denominator (LCD) to eliminate the denominators.
The LCD of the equation is the square root of x (√x). We multiply each term in the equation by the LCD:
(3√x - 2 / √x + 5) * √x = 0 * √x
Simplifying this further, we get:
3x - 2 + 5√x = 0
Step 5: Move all terms to one side of the equation.
By bringing all terms to the left side, we have:
3x + 5√x - 2 = 0
Step 6: Solve the quadratic equation by substituting a variable.
Let's substitute a variable to make it easier to solve. We can use let y = √x.
So we can rewrite the equation as:
3y^2 + 5y - 2 = 0
Step 7: Solve the quadratic equation.
Using the quadratic formula, we have:
y = (-b ± √(b^2 - 4ac)) / 2a
where a = 3, b = 5, and c = -2.
Substituting these values into the formula, we get:
y = (-5 ± √(5^2 - 4 * 3 * -2)) / (2 * 3)
Simplifying further:
y = (-5 ± √(25 + 24)) / 6
y = (-5 ± √49) / 6
Taking the square root:
y = (-5 ± 7) / 6
So we have two possible values for y:
y1 = (-5 + 7) / 6 = 1/3
y2 = (-5 - 7) / 6 = -2
Step 8: Substitute back to find x.
Since we substituted y = √x, we can substitute the values we found for y back into the equation to find the corresponding values of x.
Substituting y = 1/3:
√x = 1/3
x = (√x)^2 = (1/3)^2 = 1/9
Substituting y = -2:
√x = -2
x = (√x)^2 = (-2)^2 = 4
Therefore, the solutions to the equation 3x^(1/2) + 5 - 2x^(-1/2) = 0 are x = 1/9 and x = 4.