Question

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

Answer 1

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}`