Question
3x-1 over x+3 plus 3 equals 4x over x-3
Answers
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solve for x
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Answered by
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.
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.
Answered by
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.
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.
Answered by
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.
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.
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