How the quadratic equation solver works
The method, step by step
Step 1: compute the discriminant, D = b² − 4ac. Its sign tells you what kind of roots to expect. Step 2: apply the quadratic formula, x = (−b ± √D) ÷ 2a, once with the plus sign and once with the minus. Step 3: find the vertex at x = −b ÷ 2a and substitute back for its y-value.
Checking your answer
Two useful checks: the roots should add up to −b ÷ a, and their product should equal c ÷ a. For x² − 5x + 6, the roots 2 and 3 add to 5 and multiply to 6 — exactly −b/a and c/a. If your roots fail these checks, a sign was dropped somewhere.
Worked example
x² − 5x + 6 = 0 (a = 1, b = −5, c = 6): D = 25 − 24 = 1, which is positive, so two real roots. x = (5 ± 1) ÷ 2, giving x = 3 and x = 2. The vertex sits at (2.5, −0.25) — the bottom of the upward-opening parabola.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) / 2a. The ± means there are usually two answers: one using the plus sign and one using the minus sign.
What does the discriminant tell me?
The discriminant D = b² − 4ac decides the kind of answer before you finish solving: D > 0 means two distinct real roots, D = 0 means one repeated root, and D < 0 means no real roots (the parabola never touches the x-axis).
What is the vertex and why does it matter?
The vertex is the turning point of the parabola, at x = −b / 2a. It is the maximum or minimum value of the quadratic — useful for optimisation problems like maximum area or minimum cost.
How do I enter an equation like 2x² = 8x − 6?
Move everything to one side first: 2x² − 8x + 6 = 0, so a = 2, b = −8, c = 6. Watch the signs — a term that crosses the equals sign flips its sign.
Why did I get two answers when my homework expects one?
Most quadratics genuinely have two roots. Check whether the context rules one out — a negative length or a time before the start is often the one to discard.
More free study tools
Final Grade Calculator
See exactly what score you need on the final exam to finish the course with the grade you want.
Weighted Grade Calculator
Add each assignment with its score and weight to see your current course grade as a percentage and a letter.
Result & Grade Checker
Type in your own subject marks to get your total, percentage, letter grades, and GPA points — checked by you, stored nowhere.
Attendance Calculator
Find your attendance percentage and how many classes you must attend in a row — or can safely miss — to hit your target.
CGPA to Percentage Converter
Convert a 10-point CGPA to a percentage with the ×9.5 formula, and move between a 4.0 GPA and percentage with a standard table.
Fraction Calculator
Add, subtract, multiply, and divide fractions, with the answer simplified, as a mixed number, and as a decimal.
Percentage Tools for Marks
Quick percentage help for schoolwork: score change between two tests, average of several scores, and what percent one mark is of another.
Study Plan Calculator
Turn a pile of chapters and pages into a calm day-by-day plan that fits the days and hours you actually have.
School Age Calculator
Check a child's age on the school cutoff date and the likely kindergarten or Grade 1 entry year. Informational only — districts vary.