6 answers
(click or scroll down)
5/x^2-7x+12 + 2/x-3 = 5/x-4
Bot
answered
To solve the equation 5/x^2-7x+12 + 2/x-3 = 5/x-4:
First, factor the denominator of the first fraction: x^2-7x+12 = (x-3)(x-4)
Second, rewrite the equation with a common denominator:
5/(x-3)(x-4) + 2/(x-3) = 5/(x-4)
Third, simplify the left side using the common denominator:
(5+2(x-4))/(x-3)(x-4) = 5/(x-4)
Fourth, multiply both sides by (x-4) to eliminate the denominator on the right side:
5+2(x-4) = 5(x-3)
Fifth, simplify and solve for x:
5+2x-8 = 5x-15
-3x = -2
x = 2/3
Therefore, the solution to the equation is x = 2/3.
I hate school
answered
Check the solution
Bot
answered
To check the solution, we need to substitute x = 2/3 into the original equation and see if both sides are equal.
5/x^2-7x+12 + 2/x-3 = 5/x-4
Substituting x = 2/3, we get:
5/(2/3)^2 - 7(2/3) + 12 + 2/(2/3 - 3) = 5/(2/3 - 4)
Simplifying, we get:
5/(4/9) - 14/3 + 12 + 2/(-2 1/3) = 5/(-10/3)
20/4 - 14/3 + 12 - 6/7 = -15/2
5 - 56/21 + 252/21 - 18/7 = -15/2
35/21 - 56/21 = -15/2
-21/21 = -15/2 (not true)
Thus, the solution x = 2/3 does not make the equation true, which means there is no solution to the equation.
Tammy
answered
If I were a student posting math questions on here, I would not trust the
bot to get you the correct answer. e.g. this one .....
5/(x^2-7x+12) + 2/(x-3) = 5/(x-4) , where x ≠3,4
5/((x-3)(x-4)) + 2/(x-3) = 5/(x-4)
multiply each side by (x-3)(x-4)
5 + 2(x-4) = 5(x-3)
5 + 2x - 8 = 5x - 15
12 = 3x
x = 4
but x ≠4, so there is no solution
(but not for the reason that the bot stated , since it couldn't solve the equation properly)
Bot
answered
Thank you for bringing this to my attention. As an AI language model, my responses are based on pre-existing algorithms and statistical patterns in data. However, I strive to provide accurate and helpful responses to the best of my abilities. I appreciate any feedback or corrections to ensure that I am providing the most reliable assistance possible.
Explain Bot
answered
To solve this equation, we'll need to find the common denominator for all the fractions and then simplify the equation. Let's break it down step by step.
Step 1: Find the common denominator.
The denominators in this equation are x^2 - 7x + 12, x - 3, and x - 4. To find the common denominator, we need to factor each denominator and see what factors they have in common.
The factors of x^2 - 7x + 12 are (x - 3) and (x - 4).
The factor of x - 3 is (x - 3).
The factor of x - 4 is (x - 4).
Since all three denominators share the factors (x - 3) and (x - 4), the common denominator is (x - 3)(x - 4).
Step 2: Rewrite the fractions with the common denominator.
Now that we have the common denominator, we need to rewrite each fraction in terms of this denominator. To do this, we multiply the numerator and the denominator of each fraction by the factors that are missing in their denominators.
The first fraction is 5/(x^2 - 7x + 12). To rewrite this fraction with the common denominator, we multiply both the numerator and the denominator by (x - 3)(x - 4), which gives us (5(x - 3)(x - 4))/((x - 3)(x - 4)).
The second fraction is 2/(x - 3). To rewrite this fraction with the common denominator, we multiply both the numerator and the denominator by (x - 4), which gives us (2(x - 4))/((x - 3)(x - 4)).
Step 3: Simplify the equation.
Now that we have rewritten all the fractions with the common denominator, we can combine them into a single fraction and simplify the equation.
Combining the fractions, we have:
(5(x - 3)(x - 4) + 2(x - 4))/((x - 3)(x - 4)) = 5/(x - 4).
Expanding and simplifying the numerator, we get:
(5x^2 - 38x + 60 + 2x - 8)/((x - 3)(x - 4)) = 5/(x - 4).
Combining like terms in the numerator, we have:
(5x^2 - 36x + 52)/((x - 3)(x - 4)) = 5/(x - 4).
We can now cancel out the common factors in the numerator and denominator:
5x^2 - 36x + 52 = 5(x - 3).
Expanding the right side, we get:
5x^2 - 36x + 52 = 5x - 15.
Step 4: Solve the equation.
Now that we have a simplified equation, we can solve it by moving all terms to one side and setting the equation equal to zero.
Subtracting 5x and 52 from both sides, we get:
5x^2 - 41x - 67 = 0.
This is now a quadratic equation. To solve it, you can use factoring, completing the square, or the quadratic formula.