How do you solve #\frac { x ( x + 1) } { 3} - \frac { x - 1} { 2} = x + x ^ { 2}#?

1 Answer
Ben
Jun 26, 2018

Re-organize the equation into a quadratic equal to 0, then solve using the quadratic formula. You will find that #x=-(7+-sqrt(97))/8# or #x~={-2.106107225,0.356107225}#

Explanation:

First, we'll simplify the fractions. The common factor between 3 and 2 is 6, so we will multiply both sides to eliminate the denominators:

#6((x(x+1))/3-(x-1)/2)=6(x+x^2)#

#2x(x+1)-3(x-1)=6(x^2+x)#

Expand and distribute all factors:

#2x^2+2x-3x+3=6x^2+6x#

Move all terms to one side of the equation:

#2x^2-x+3=6x^2+6x#

#-4x^2-7x+3=0#

Now, we have a quadratic equation that we can apply The Quadratic Formula to:

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

#a=-4#

#b=-7#

#c=3#

#x=(-(-7)+-sqrt((-7)^2-4(-4)(3)))/(2(-4))#

#x=(7+-sqrt(49-(-48)))/(-8)#

#color(green)(x=(7+-sqrt(97))/(-8)#

The above is the simplest we can go with integers (97 is not a perfect square), but if we want to write it as a decimal we can do it as follows:

#x=-7/8+-sqrt(97)/(-8)#

#x~=-0.875+-9.848857802/(-8)#

#x~=-0.875+-(-1.231107225)#

#x~={-0.875+(-1.231107225),-0.875-(-1.231107225)}#

#color(green)(x~={-2.106107225,0.356107225}#

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