^{2 }- x - 14 over x-2 divided by 4x

^{2 }+ x-14 over x+2

1. 1 over t-1 minus 3 over t-3 both equal 1/4

2. 4x^{2 }- x - 14 over x-2 divided by 4x^{2 }+ x-14 over x+2

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4x^{2} - x - 14 ÷ 4x^{2} + x - 14

x - 2 x + 2

dividing by a fraction is the same a multiplying by the reciprocal of the fraction.

So we will change this to multiplication, and factor the quadratics

(4x + 7)(x - 2) (4x - 7)(x + 2)

(x - 2) (x + 2)

(4x + 7)(4x - 7)

16x^{2} - 49

Nkeiruka:

1. I take your "both equal" as follows

1 / (t - 1) - 3 / (t - 3) = 1 / 4.

The LCM on the left is (t-1)(t-3). The equation is

[t - 3 - 3(t - 1)] / (t-1)(t-3) = 1 / 4. Never divide by zero, so assume that t is not equal to 1 or t is not equal to 3, or both.

Now simplify the above expression as

[t - 3 - 3t +3 ] / (t-1)(t-3) = 1 / 4

Cross-multiply and multiply out the factors.

- 8t = t^{2} - 4t + 3

You get

t^{2} + 4t + 3 = 0.

Factorize.

(t +1) ( t + 3) = 0.

So the roots are either t = -1 or t = -3.

Your Number 2 is NOT an equation, it is just an expression, not equated to anything. Incomplete?

Dattaprabhakar (Dr. G.)

Thanks so much Doctor!!!

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