All Junior Cert Mathematics posts

'Solve the equation x-2/x+1 - 5/x-1 = 1/3..... Write your answer in the form p+/- q (square root) r, where p, q and r are natural numbers'

x-2/x+1 - 5/x-1 was the last question, and the answer was x(squared)-8x-3/x(squared)-1.

I checked the answers, and the answer is x = 6+/-2(square root)10. Id really appreciate some help, thanks.

1. #### Dazzla16

Hi maria! Here are the steps involved in solving that equation.

[(x-2)/(x+1)] - [5/(x-1)] = 1/3

Multiply both sides by 3:

[3(x-2)/(x+1)] - [3(5)/(x-1)] = 3(1/3)

[(3x-6)/(x+1)] - [15/(x-1)] = 1

Now you have to express the two fractions in the same denominator. To do this, multiply the numerator and the denominator of (3x-6)/(x+1) by (x-1) and multiply the numerator and the denominator of 15/(x-1) by (x+1). You get:

[(3x^2 - 9x + 6)/(x^2 - 1)] - [(15x + 15)/(x^2 - 1)] = 1

You can subtract the two fractions from each other since they are expressed in the same denominator.

[(3x^2 - 24x - 9)/(x^2 - 1)] = 1

Then, express 1 as a fraction.

[(3x^2 - 24x - 9)/(x^2 - 1)] = 1/1

Cross multiply and you get:

3x^2 - 24x - 9 = x^2 - 1

Make one side equal to 0:

2x^2 - 24x - 8 = 0

Divide both sides by 2:

x^2 - 12x - 4 = 0

Now use the quadratic formula. Remember to make a = 1, b = -12 and c = -4.

x = [-b +/- sqrt(b^2 - 4ac)]/2a

x = [-(-12) +/- sqrt((-12)^2 - 4(1)(-4))]/2(1)

x = [12 +/- sqrt(160)]/2

Square root of 160 is simplified to 4sqrt(10).

x = [12 +/- 4sqrt(10)]/2

x = 6 +/- 2sqrt(10)

Hope it helps :)

2. #### ismisemaria

Hey, thanks that's a great help, just confused on what do you mean by 'sqrt of 160 is simplified to 4sqrt(10)', how is it simplified, as we haven't covered that in class yet

3. #### Dazzla16

Hi again. The simplification of surds is covered in the section on Indices. In order to simplify a number in surd form, you have to see if the surd can have one factor being a perfect square (such as 4, 9, 16, 25, etc.).

In our case, sqrt(160) can have a factor being a perfect square, which is sqrt(16).

sqrt(160) = sqrt(16) X sqrt(10)

We already know that the square root of 16 is equals to 4, so we replace sqrt(16) with 4.

sqrt(160) = 4 X sqrt(10)

sqrt(160) = 4sqrt(10)

Hope this helps :)

4. #### ismisemaria

Seriously thank you so much! This really helped!