The solutions to the general-form quadratic function can be given by a simple equation called the quadratic equation. To solve this equation, recall the completed square form of the quadratic equation derived in the previous section:
In this case, since we're looking for the root of this function. To solve, first subtract c and divide by a:
Take the (plus and minus) square root of both sides to obtain:
Subtracting from both sides:
This is the solution but it's in an inconvenient form. Let's rationalize the denominator of the square root:
Now, adding the fractions, the final version of the quadratic formula is:
This formula is very useful, and it is suggested that the students memorize it as soon as they can.
The part under the radical sign, , is called the discriminant, Δ. The value of the discriminant tells us some useful information about the roots.
- If Δ > 0, there are two unique real solutions.
- If Δ = 0, there is one unique real solution.
- If Δ < 0, there are two unique, conjugate imaginary solutions.
- If Δ is a perfect square then the two solutions are rational, otherwise they are irrational conjugates.
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