How do you simplify \frac { 4} { 3x - 6} - \frac { 5} { 4x + 4} + \frac { 2} { 5x + 10}43x654x+4+25x+10?

1 Answer
Sep 20, 2017

(29x^2+216x+412)/((4x+4)(5x+10)(3x-6))29x2+216x+412(4x+4)(5x+10)(3x6)

Explanation:

You first have to make every denominator the same, so multiply all the fractions by another fraction that equals one.

=4/(3x-6)*((4x+4)(5x+10))/((4x+4)(5x+10)) - 5/(4x+4)((3x-6)(5x+10))/((3x-6)(5x+10))+2/(5x+10) ((3x-6)(4x+4))/((3x-6)(4x+4))=43x6(4x+4)(5x+10)(4x+4)(5x+10)54x+4(3x6)(5x+10)(3x6)(5x+10)+25x+10(3x6)(4x+4)(3x6)(4x+4)

= (4(4x+4)(5x+10)-5(5x+10)(3x-6)+2(3x-6)(4x+4))/((4x+4)(5x+10)(3x-6))=4(4x+4)(5x+10)5(5x+10)(3x6)+2(3x6)(4x+4)(4x+4)(5x+10)(3x6)

You get a really large fraction, but from here you just expand simplify. You don't have to expand the denominator because nothing factors easily yet.

=(4(20x^2+60x+40)-5(15x^2-60)+2(12x^2-12x-24))/((4x+4)(5x+10)(3x-6))=4(20x2+60x+40)5(15x260)+2(12x212x24)(4x+4)(5x+10)(3x6)
=(80x^2+240x+160-75x^2+300+24x^2-24x-48)/((4x+4)(5x+10)(3x-6)=80x2+240x+16075x2+300+24x224x48(4x+4)(5x+10)(3x6)
=(29x^2+216x+412)/((4x+4)(5x+10)(3x-6))=29x2+216x+412(4x+4)(5x+10)(3x6)

Using the quadratic formula, we can tell what the roots are for the quadratic in the numerator.

x=(-b+-sqrt(b^2-4ac))/(2a) = (-216+-sqrt(216^2-4(29)(412)))/(2(29))=(-216+-sqrt(46656-47792))/58=(-216+-sqrt(-1136))/58x=b±b24ac2a=216±21624(29)(412)2(29)=216±466564779258=216±113658

Since taking the square root of a negative number gives us imaginary answers, we can conclude that the quadratic cannot factor further, and this is the final, simplified fraction:
(29x^2+216x+412)/((4x+4)(5x+10)(3x-6))29x2+216x+412(4x+4)(5x+10)(3x6)