Question

3x-1 over x+3 plus 3 equals 4x over x-3

Answers

GPT 3.5
To solve this equation, we need to combine like terms and simplify.

First, let's break down the equation:

(3x-1)/(x+3) + 3 = 4x/(x-3)

To simplify further, we need a common denominator for the fractions on the left side of the equation.

Multiplying the first fraction by (x-3)/(x-3) and the second fraction by (x+3)/(x+3), we get:

(3(x-3) - 1(x+3))/(x+3) + 3 = 4x/(x-3)

Expanding the numerator in the first fraction, we get:

(3x - 9 - x - 3)/(x+3) + 3 = 4x/(x-3)

Combining like terms, we have:

(2x - 12)/(x + 3) + 3 = 4x/(x - 3)

To get rid of the denominators, we can multiply both sides of the equation by (x + 3)(x - 3) to clear them:

(x + 3)(x - 3)(2x - 12)/(x + 3) + 3(x + 3)(x - 3) = 4x(x + 3)(x - 3)/(x - 3)

This simplifies to:

(2x - 12)(x - 3) + 3(x + 3)(x - 3) = 4x(x + 3)

Expanding and simplifying further, we get:

2x^2 - 6x - 12x + 36 + 3x^2 - 9 = 4x^2 + 12x

Combining like terms, we have:

5x^2 - 18x + 27 = 4x^2 + 12x

Rearranging the terms and setting the equation equal to zero, we get:

5x^2 - 4x^2 - 18x - 12x + 27 = 0

x^2 - 30x + 27 = 0

Now we can solve this quadratic equation for x using methods such as factoring, completing the square, or the quadratic formula. After finding the values of x that satisfy this equation, we can substitute them back into the original equation to verify if they are the correct solutions.
solve for x
GPT 3.5
To solve the quadratic equation x^2 - 30x + 27 = 0, we can factor it if possible. Let's attempt to factor it:

x^2 - 30x + 27 = 0
(x - 3)(x - 9) = 0

Setting each factor to zero, we get:

x - 3 = 0 or x - 9 = 0

Solving for x:

x = 3 or x = 9

Therefore, the solutions to the equation x^2 - 30x + 27 = 0 are x = 3 and x = 9. To verify these solutions, we can substitute them back into the original equation and check if they satisfy it.
are you sure
GPT 3.5
Let's double-check the calculation for the quadratic equation x^2 - 30x + 27 = 0 by using the quadratic formula:

The general form of a quadratic equation is ax^2 + bx + c = 0. For our equation x^2 - 30x + 27 = 0, we have a = 1, b = -30, and c = 27.

The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values:
x = (30 ± √((-30)^2 - 4*1*27)) / 2*1
x = (30 ± √(900 - 108)) / 2
x = (30 ± √792) / 2
x = (30 ± √(4*198)) / 2
x = (30 ± 2√198) / 2
x = 15 ± √198

So, the solutions to the quadratic equation x^2 - 30x + 27 = 0 are x = 15 + √198 and x = 15 - √198.