How do you solve #3-((a-1)/(a+3)) = ((a^2-5)/(a+3))#?

Question

Eva Abramson · Accepted Answer

Multiply both sides by #(a+3)# and then solve the resulting quadratic to find #a=5#

The denominator is the same for both fractions in the equation. We can use this to our advantage by multiplying both sides by the denominator to 'clear' it from our equation:

#color(red)((a+3))(3-(a-1)/(a+3))=cancel(color(red)((a+3)))((a^2-5)/cancel(a+3))#

#color(red)((a+3))xx3-cancel(color(red)((a+3)))((a-1)/cancel(a+3))=a^2-5#

#color(red)((a+3))xx3-(a-1)=a^2-5#

Now that we've 'cleared' the denominators from the equation, we simplify all the terms:

#3a+9-(a-1)=a^2-5#

#3a+9-a+1=a^2-5#

#2a+10=a^2-5#

#-a^2+2a+10=-5#

#-a^2+2a+15=0#

Now, we have a quadratic equation with the following coefficients:

#c_1=-1# #c_2=2# #c_3=15#

I wrote the equation using #c_n# terms instead of #a# #b# and #c# to differentiate from the #a# we are trying to solve. Let's plug those into the quadratic formula:

#a=(-c_2+-sqrt(c_2^2-4c_1c_3))/(2c_1)#

#a=(-2+-sqrt(2^2-4(-1)(15)))/(2(-1))#

#a=(-2+-sqrt(4+60))/(-2)=(-2+-sqrt(64))/(-2)#

#a=(-2+-8)/(-2)#

#a={(-2+8)/(-2),(-2-8)/(-2)}#

#a={6/(-2),(-10)/(-2)}#

#a={-3,5}#

EDIT: Updating results

Finally, we'll plug the solutions for #a# back in to the original equation:

When #a=-3#:

#3-((-3)-1)/((-3)+3)=((-3)^2-5)/((-3)+3)#

#3-(-4)/(0)=(4)/(0)#

Since we have a solution where there's division by zero, this solution is considered extraneous, making it invalid for this question.

When #a=5#:

#3-((5)-1)/((5)+3)=((5)^2-5)/((5)+3)#

#3-(4)/(8)=(20)/(8)#

#3=20/8+4/8=(20+4)/8=24/8#

#3=3#

This means that the positive root is valid, and gives us our final answer:

#color(green)(a=5)#

Jameson Allen · Answer

undefined To solve the equation 3-((a-1)/(a+3)) = ((a^2-5)/(a+3)), we can follow these steps:

1. Start by multiplying both sides of the equation by (a+3) to eliminate the denominators.
2. Simplify the equation by distributing the multiplication on both sides.
3. Combine like terms and move all terms to one side of the equation.
4. Rearrange the equation to form a quadratic equation.
5. Solve the quadratic equation using factoring, completing the square, or the quadratic formula.
6. Check the solutions obtained by substituting them back into the original equation to ensure they satisfy the equation.

By following these steps, you can find the solution(s) to the given equation.