Question

Show that each algebraic fraction simplifies to the given expression.

a) 2/x+1 + 5/x+2 = 3 simplifies to 3x² + 2x - 3 = 0

b) 3/4x+1 - 4/x+2 = 2 simplifies to 8x² + 31x + 2 = 0

c) 2/2x-1 - 6/x+1 = 11 simplifies to 22x² +21x - 19 = 0

Answer 1

Hello,

a)
we suppose that x≠-1 and x≠-2
2/(x+1)+5/(x+2)=3
==>2(x+2)+5(x+1)=3(x+1)(x+2)
==>7x+9=3x²+9x+6
==>3x²+2x+3=0

b)
we suppose that x≠-1/4 and x≠-2
3/(4x+1)-4/(x+2)=2
==>3(x+2)-4(4x+1)=2(4x+1)(x+2)
==>-13x+2=8x²+18x+4
==>8x²+31x+2=0

c)
we suppose that x≠1/2 and x≠-1
2/(2x-1)-6/(x+1)=11
==>2(x+1)-6(2x-1)=11(2x-1)(x+1)
==>-10x+8=22x²+11x-11
==>22x²+21x-19=0

Answer 2

a) 2/x+1 + 5/x+2 = 3 simplifies to 3x² + 2x - 3 = 0

2(x+2) + 5(x+1) = 3(x+2)(x+1)
2x + 4 + 5x + 5 = (3x+6)(x+1)
7x + 9 = 3x² + 3x + 6x + 6
7x + 9 = 3x² + 9x + 6
0 = 3x² + 9x - 7x + 6 - 9
0 = 3x² + 2x - 3

b) 3/4x+1 - 4/x+2 = 2 simplifies to 8x² + 31x + 2 = 0

3(x+2) - 4(4x+1) = 2(x+2)(4x+1)
3x + 6 - 16x - 4 = (2x+4)(4x+1)
-13x + 2 = 8x² + 2x + 16x + 4
0 = 8x² + 18x + 13x + 4 - 2
0 = 8x² + 31x + 2

c) 2/2x-1 - 6/x+1 = 11 simplifies to 22x² +21x - 19 = 0

2(x+1) - 6(2x-1) = 11(x+1)(2x-1)
2x + 2 - 12x + 6 = (11x+11)(2x-1)
-10x + 8 = 22x² - 11x + 22x - 11
0 = 22x² + 11x + 10x - 11 - 8
0 = 22x² + 21x - 19