Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a solves every quadratic.
About this topic
The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) solves any quadratic equation ax² + bx + c = 0, whether or not it factors. Identify a, b and c, substitute carefully, and simplify. The ± gives up to two solutions.
It comes from completing the square on the general equation. For 2x² + 3x − 5 = 0, a = 2, b = 3, c = −5, so the formula gives (−3 ± √(9 + 40)) / 4 = (−3 ± 7)/4, which is 1 or −5/2. When the root is not a perfect square, as in x² + 2x − 1 = 0, the answers are −1 ± √2, which you can leave exact or round.
It is the reliable fallback when factoring fails, and it is used in physics for projectile timing, in engineering for design problems and in finance for rates. Calculators and programs use exactly this formula.
The common errors are forgetting that the whole numerator is divided by 2a, writing −b incorrectly when b is already negative (the opposite of −5 is 5), and mishandling b² − 4ac when c is negative, since subtracting a negative adds. Put the equation in standard form before reading a, b and c.
Key ideas
- Identify a, b and c
- Compute the discriminant first
- Gives two, one or no real roots
Quick check
Use cases: where Quadratic Formula helps
- Homework: review the key idea behind Quadratic Formula before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Quadratic Formula without notes.
- Visual learning: play with the interactive lab to see how Quadratic Formula changes when you adjust its inputs.
- Teaching a friend: explain Quadratic Formula in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Quadratic Formula?
x = (−b ± √(b² − 4ac)) / 2a solves every quadratic.
What will I learn in Quadratic Formula?
Identify a, b and c. Compute the discriminant first. Gives two, one or no real roots.
Is Quadratic Formula part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Quadratic Formula lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Quadratic Formula?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.