How do you solve #1/(x-4) + (x-4)/(x-4) = 7 / ( x^2+x-20)#?

Answer 1

Eliminate the denominators by multiplying both sides by #(x-4)(x+5)#.
Solve the resulting quadratic.

Given: #1/(x-4) + (x-4)/(x-4) = 7 / ( x^2+x-20)#
Eliminate the denominators by multiplying both sides by #(x-4)(x+5)#.
#x+5+(x-4)(x+5) = 7#
#x+5 + x^2+x-20 = 7#
#x^2+2x-22 = 0#

Use the quadratic formula:

#x = (-2+-sqrt(2^2-4(1)(-22)))/(2(1))#
#x = (-2+-2sqrt(23))/2#
#x = -1+sqrt(23) and x = -1-sqrt(23)#

This agrees with WolframAlpha

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Answer 2

To solve the equation 1/(x-4) + (x-4)/(x-4) = 7/(x^2+x-20), we can simplify the equation by combining the fractions with a common denominator. The common denominator is (x-4)(x+5). Simplifying the equation, we get (x+5 + (x-4))/(x-4)(x+5) = 7/(x^2+x-20). Combining like terms in the numerator, we have (2x+1)/(x-4)(x+5) = 7/(x^2+x-20). Cross-multiplying, we get (2x+1)(x^2+x-20) = 7(x-4)(x+5). Expanding and simplifying the equation, we have 2x^3 + 3x^2 - 39x - 20 = 7x^2 - 7x - 140. Rearranging the terms, we get 2x^3 - 4x^2 + 32x + 120 = 0. Factoring out 2, we have 2(x^3 - 2x^2 + 16x + 60) = 0. Factoring the cubic polynomial, we find (x+2)(x-6)(x-5) = 0. Therefore, the solutions to the equation are x = -2, x = 6, and x = 5.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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