Question

How can you combine the numerators on the right side of the equation and place everything over 2a, considering that the square root in the numerator is both added to and subtracted from -b? Additionally, explain the concept of having two possible solutions.

Answer 1

The ± in the quadratic formula accounts for the fact that a quadratic equation can have two solutions (real or complex) unless the discriminant is zero, in which case there is only one real solution.

How is it so?

Having two solutions (real or complex) could be found in solving a quadratic equation and expressing the solutions in a specific form. The general form of a quadratic equation is:

`\[ax^2 + bx + c = 0\]`

To solve for
`\(x\)`, one common method is completing the square. The quadratic formula is derived from completing the square, and it is given by:

`\[x = (-b \pm √(b^2 - 4ac))/(2a)\]`

It is possible to combine the numerators on the right side and place everything over
`\(2a\)` as thus:

`\[x = (-b \pm √(b^2 - 4ac))/(2a)\]`

Combine the numerators:

`\[x = (-b \pm √(b^2 - 4ac))/(2a) * (1)/(1)\]`

This gives:

`\[x = (-b \pm √(b^2 - 4ac))/(2a) * (1)/(1) \\= (-b \pm √(b^2 - 4ac))/(2a)\]`

So, there's no need to explicitly combine the numerators; the expression is already in a simplified form.

Examining the concept of having two possible solutions, it is tied to the ± symbol in the quadratic formula. The term
`\(√(b^2 - 4ac)\)` inside the square root can either be positive or negative (or zero). This leads to two possible solutions for
`\(x\)`: one with the positive square root (the
`\(+\)` case) and one with the negative square root (the
`\(-\)` case).

If the discriminant
`(\(b^2 - 4ac\))` is positive, there are two distinct real solutions.

If the discriminant is zero, there is one real solution (both
`\(+\)` and
`\(-\)` cases result in the same value).

If the discriminant is negative, there are two complex conjugate solutions (involving imaginary numbers).

Finally, the ± in the quadratic formula accounts for the fact that a quadratic equation can have two solutions (real or complex) unless the discriminant is zero, in which case there is only one real solution.