Question

What is an expression for the sum of the roots of quadratic ax^2 + bx^2 + c?

Answer 1

`x_1 + x_2 = -b/a`

Explanation:

We know by the quadratic formula that

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

So our two solutions will be

`x_1 = (-b + sqrt(b^2 - 4ac))/(2a)`
`x_2 = (-b - sqrt(b^2 - 4ac))/(2a)`

Therefore, the sum will give

`x_1 + x_2 = (-b + sqrt(b^2 - 4ac))/(2a) + (-b - sqrt(b^2 - 4ac))/(2a)`

`x_1+ x_2 = (-b - b + sqrt(b^2 - 4ac) - sqrt(b^2 - 4ac))/(2a)`

`x_1 + x_2 = (-2b)/(2a)`

`x_1 + x_2 = -b/a`

Let's try a few easy examples. In the equation `x^2 + 5x + 6 = 0`, we have roots `x = -3` and `x= -2`. The sum is `-3 + (-2) = -5`. Using the above formula, we get

`x_1 + x_2 = -5/1 = -5`

Which is the same result we got if we manually added them.

For another example, we can use `x^2 - 1= 0`. Here, `x = +1` and `x = -1`. Therefore,

`x_1 + x_2 = +1 + (-1) = 0`

There is no `x` term in the equation, so `b` will clearly be `0`.

`x_1 + x_2 = 0/1 = 0`

This formula will clearly not work for non-quadratic equations (that's to say there needs to be a term of degree `2`, and the degree `2` term must be the maximum degree of the equation, or else the formula won't function properly).

Hopefully this helps!