Question

How to solve 2x-1/3 = x^2+2x/5 using the quadratic formula?

Answer 1

`x=(12+-sqrt69)/15`

Explanation:

I am interpreting the given equation as:

`2x-1/3=x^2+(2x)/5`

Move these all to the same side of the equation.

`0=x^2+(2x)/5-2x+1/3`

Simplify the terms with `x`.

`0=x^2+(2x)/5-(10x)/5+1/3`

`0=x^2-(8x)/5+1/3`

We could apply the quadratic formula to this, but it would be simpler if we eliminated the fractions. To do this, multiply both sides by `15`.

`0=15x^2-24x+5`

We can now use the quadratic formula, which states that for a quadratic function `0=ax^2+bx+c`, the roots are

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

Thus, `a=15,b=-24,c=5`, so

`x=(-(-24)+-sqrt((-24)^2-(4xx15xx5)))/(2xx15)`

`x=(24+-sqrt(576-300))/30`

`x=(24+-sqrt276)/30`

`x=(24+-sqrt(2^2xx69))/30`

`x=(24+-2sqrt69)/30`

`x=(12+-sqrt69)/15`