Help Pleasee!
Rational Exponet equations
1. 3x^5/2- 9 = 0
2. (2x+3)^1/2=4
Equations With rational expressions:
1. 3/x+3 - 1/x-2 = 5/2x+6
Sure, I can help you with those equations. Let's go through each equation step by step.
1. 3x^(5/2) - 9 = 0
To solve this equation, we need to isolate the variable x. Let's start by adding 9 to both sides of the equation:
3x^(5/2) = 9
Next, we need to eliminate the exponent of 5/2. We can do this by raising both sides of the equation to the reciprocal of the exponent:
(3x^(5/2))^(2/5) = 9^(2/5)
Simplifying this gives us:
3^(2/5) * (x^(5/2))^(2/5) = 9^(2/5)
Now, let's simplify the left side of the equation:
3^(2/5) * x = 9^(2/5)
To solve for x, we can divide both sides of the equation by 3^(2/5):
x = (9^(2/5)) / (3^(2/5))
Using a calculator, we can find the numerical value of x.
2. (2x+3)^(1/2) = 4
Similar to the previous equation, we need to isolate the variable x. Let's start by squaring both sides of the equation:
[(2x+3)^(1/2)]^2 = 4^2
Simplifying this gives us:
2x + 3 = 16
Next, let's isolate x by subtracting 3 from both sides of the equation:
2x = 16 - 3
Simplifying further gives us:
2x = 13
Finally, we divide both sides of the equation by 2 to solve for x:
x = 13/2
Now let's move on to the equation with rational expressions.
1. 3/(x+3) - 1/(x-2) = 5/(2x+6)
To solve this equation, we can start by clearing the denominators. We can do this by multiplying every term in the equation by the least common denominator (LCD), which in this case is (x+3)(x-2)(2x+6).
(x+3)(x-2)(2x+6)[3/(x+3) - 1/(x-2)] = (x+3)(x-2)(2x+6)(5/(2x+6))
Simplifying each term gives us:
3(x-2)(2x+6) - (x+3)(2x+6) = 5(x+3)(x-2)
Expand and simplify each term:
6x^2 + 18x - 12x - 36 - 2x^2 - 6x - 6 = 5(x^2 - 2x + 3x - 6)
Combine like terms:
6x^2 + 6x - 36 - 2x^2 - 6x - 6 = 5x^2 + 5x - 15
Rearrange terms to get all terms on one side:
6x^2 - 2x^2 + 6x + 6x - 5x^2 - 5x + 36 + 6 + 15 = 0
Combine like terms again:
- x^2 + 13x + 57 = 0
Now, we have a quadratic equation. To solve for x, we can use factoring, completing the square, or the quadratic formula. In this case, factoring might be challenging, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = -1, b = 13, and c = 57. Plugging in these values into the quadratic formula will give us the solutions for x.
I hope this explanation helps!