Question

If both of the zeros of `ax^2+bx+c` are positive or both negative then what can you say about the signs of `a, b` and `c` ?

Answer 1

See explanation...

Explanation:

Given a quadratic polynomial:

`ax^2+bx+c`

• If the zeros are both positive then `a` and `c` have the same sign and `b` has the opposite sign.

• If the zeros are both negative then `a`, `b` and `c` have the same sign.

To see these, suppose the zeros are `x=alpha` and `x=beta`.

We have:

`ax^2+bx+c = a(x-alpha)(x-beta)`

`color(white)(ax^2+bx+c) = a(x^2-(alpha+beta)x+alphabeta)`

`color(white)(ax^2+bx+c) = ax^2-(alpha+beta)ax+alphabetaa`

Matching the coefficients, we find:

`{ (b = -(alpha+beta)a), (c = alphabetaa) :}`

If `alpha > 0` and `beta > 0` then `-(alpha+beta) < 0` and `alphabeta > 0`. Hence `b` has the opposite sign to `a` and `c` has the same sign as `a`.

If `alpha < 0` and `beta < 0` then `-(alpha+beta) > 0` and `alphabeta > 0`. Hence `a`, `b` and `c` all have the same sign.

`color(white)()`
Note

The converses of these two results are not generally true.

There are quadratic equations `ax^2+bx+c` with `a, c` having the same sign and `b` having the same or opposite sign which have no real zeros.

For example:

`x^2+x+1`

has no real zeros.

It still has two zeros, but they are so called "imaginary" a.k.a. non-real complex numbers. They do not lie on the real number line, but somewhere else in the complex plane. The real number line forms the `x` axis of the complex plane, the `y` axis being called the imaginary axis.

All complex numbers can be expressed in the form `x+yi`, where `x, y` are real numbers and `i` is the imaginary unit, satisfying:

`i^2 = -1`

For example, the zeros of `x^2+x+1` are:

`-1/2+sqrt(3)/2i" "` and `" "-1/2-sqrt(3)/2i`

It is unfortunate that the term imaginary is used of such numbers - coined when such numbers were poorly understood. They are just as real as Real numbers and better behaved in some respects.

Answer 2

Some more explanation...

Explanation:

Attempting to make this a little easier to follow, let us start with the simpler case where the leading coefficient is `1`

`x^2+px+q`

If `alpha` and `beta` are the two zeros, then both `(x-alpha)` and `(x-beta)` must be factors.

Using FOIL to help multiply, we find:

`(x-alpha)(x-beta) = overbrace((x * x))^"First"+overbrace(x * (-beta))^"Outside"+overbrace((-alpha) * x)^"Inside"+overbrace((-alpha) *(-beta))^"Last"`

`color(white)((x-alpha)(x-beta)) = x^2-betax-alphax+alphabeta`

`color(white)((x-alpha)(x-beta)) = x^2-(alpha+beta)x+alphabeta`

Comparing this with the expression `x^2+px+q` we find:

• If `alpha > 0` and `beta > 0` then `-(alpha+beta) < 0`, so `p < 0`.
  Also `alphabeta > 0`, so `q > 0`.`color(white)(0/0)`
  In summary, the pattern of the signs of the coefficients of `x^2+px+q` is `+ - +`.`color(white)(0/0)`
  For example `x^2-3x+2 = (x-1)(x-2)` has zeros `1` and `2`.`color(white)(0/0)`

• If `alpha < 0` and `beta < 0` then `-(alpha+beta) > 0`, so `p > 0`.`color(white)(0/0)`
  Also `alphabeta > 0`, so `q > 0`.`color(white)(0/0)`
  In summary, the pattern of the signs of the coefficients of `x^2+px+q` is `+ + +`.`color(white)(0/0)`
  For example `x^2+3x+2 = (x+1)(x+2)` has zeros `-1` and `-2`.

The more general case of `ax^2+bx+c` is similar, but we have to deal with that initial coefficient `a`, which can be positive or negative.

We have:

`ax^2+bx+c = a(x^2+b/ax+c/a) = a(x-alpha)(x-beta)`

• If `alpha > 0` and `beta > 0` then the coefficients `1`, `b/a` and `c/a` have signs in the pattern `+ - +`.`color(white)(0/0)`
  Hence if `a > 0` then the pattern of the signs of the coefficients of `ax^2+bx+c` is also `+ - +`.`color(white)(0/0)`
  If instead `a < 0` then the pattern of the signed of the coefficients of `ax^2+bx+c` is `- + -`.`color(white)(0/0)`

• If `alpha < 0` and `beta < 0` then the coefficients `1`, `b/a` and `c/a` have signs in the pattern `+ + +`.`color(white)(0/0)`
  Hence if `a > 0` then the pattern of the signs of the coefficients of `ax^2+bx+c` is also `+ + +`.`color(white)(0/0)`
  If instead `a < 0` then the pattern of the signs of the coefficients of `ax^2+bx+c` is `- - -`.