Quadratic equations and discriminants

The coefficients of a quadratic encode its roots and graph. Factoring, completing the square and the quadratic formula provide different ways to reveal that information.

A reliable method

Set the equation equal to zero. For ax² + bx + c, examine b² − 4ac before deciding whether there are two, one or no distinct real solutions.

The common trap

A zero discriminant gives one distinct real root with multiplicity two. It does not mean the root itself is zero.

Check and transfer

The roots sum to −b/a and multiply to c/a. Use these relationships when the requested quantity is a sum or product rather than solving for every root separately.

Worked comparison: identify the structure

For 2x²−6x+4=0, roots 1 and 2 have sum 3 and product 2. The coefficient shortcuts −b/a and c/a reproduce those quantities. Omitting the leading coefficient gives 6 and 4, neither the requested sum nor product.

Transfer and explain

For x²−8x+c=0 to have a repeated real root, set 64−4c=0, so c=16. The repeated root is 4, illustrating why c and the root itself must not be confused.

WORKED EXAMPLE

What is the sum of the roots of 3x² − 18x + 2 = 0?

For ax² + bx + c = 0, the root sum is −b/a.
Use the signed coefficient b rather than only its magnitude.
The requested value is 6. Substitute or reverse the operation to check this result.

Answer: 6

Compare the other answer choices

Divide by the leading coefficient twice.

2/3

Report the root product c/a instead of the sum.

-6

Use b/a instead of negative b/a.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

CLEP College Algebra exam overview Siyavula Grade 10 foundation reference