Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula. Get real and complex roots.
1x² + -5x + 6 = 0
Discriminant
1
Type: two real
Root 1
3.000
Root 2
2.000
Vertex
(2.5, -0.25)
Last updated: February 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
A student in Lucknow needs to solve x squared minus 5x plus 6 equals 0. Entering a equal to 1, b equal to minus 5 and c equal to 6, the calculator returns x equal to 2 and x equal to 3. Substituting either value back verifies the answer: 4 minus 10 plus 6 is 0, and 9 minus 15 plus 6 is 0. The discriminant, b squared minus 4ac, is 25 minus 24, or 1, a positive perfect square, which is why the two roots are real and rational.
How to Use This Calculator
- Step 1: Write the equation in standard form ax squared plus bx plus c equals 0 with all terms on one side.
- Step 2: Enter the coefficients a, b and c in the three fields.
- Step 3: Choose whether you want real roots only or complex roots displayed.
- Step 4: Read the two roots, and check the sign of the discriminant shown to understand the nature of the solutions.
Worked Example
A student in Lucknow needs to solve x squared minus 5x plus 6 equals 0. Entering a equal to 1, b equal to minus 5 and c equal to 6, the calculator returns x equal to 2 and x equal to 3. Substituting either value back verifies the answer: 4 minus 10 plus 6 is 0, and 9 minus 15 plus 6 is 0. The discriminant, b squared minus 4ac, is 25 minus 24, or 1, a positive perfect square, which is why the two roots are real and rational.
Tips and Common Mistakes
- •Tip 1: If a is 0, the equation is not quadratic; solve it as a linear equation instead.
- •Tip 2: When the discriminant is negative, the roots are complex and come as a conjugate pair, so allow complex output to see them.
- •Tip 3: Move every term to one side before entering values so the equation matches the standard form.
- ✗Mistake 1: Entering the sign of b incorrectly; x squared minus 5x plus 6 has b equal to -5, not 5.
- ✗Mistake 2: Assuming a zero discriminant means no roots; it means the two roots are equal, not absent.
Frequently Asked Questions
What is the quadratic formula?
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 5x + 6 = 0 the calculator applies it and returns x = 2 and x = 3.
What does the discriminant tell me?
The discriminant b² − 4ac decides the nature of the roots: positive means two real roots, zero means one repeated real root, and negative means two complex roots. For x² + 2x + 5 = 0 the discriminant is −16, so the roots are complex.
Why can't I solve when a = 0?
With a = 0 the equation becomes linear, bx + c = 0, and the formula would require dividing by zero. The calculator flags this and asks you to solve it as a linear equation instead.
Where do quadratic equations appear in Indian exams?
Quadratic equations are a full chapter in Class 10 CBSE and state board mathematics, and they also appear in JEE, NDA and other competitive tests. The calculator is equally handy for physics problems such as projectile motion.
How do I check that my roots are correct?
Substitute each root back into the original equation — it must equal zero. For x = 2 in x² − 5x + 6 = 0, 4 − 10 + 6 = 0. The calculator shows the substituted working so you can verify step by step.
Compare these tools
- Percentage Calculator — Calculate percentages, percentage difference and percentage change instantly
- Rounding Calculator — Round any number to the nearest value, or up and down, to any decimal places
- Standard Form Calculator — Convert numbers to and from standard form (scientific notation) with one click
You might also need
Related calculators from other categories
Science & Weather
Ideal Gas Law Calculator
Solve PV = nRT for pressure, volume, moles or temperature with the gas constant R = 8.314. Enter any three to find the fourth.
Science & Weather
Acceleration Calculator
Calculate acceleration from initial and final velocity over time, plus the distance covered. Handles speeding up and braking.
Health & Fitness
Army Body Fat Calculator
Estimate body fat percentage with the US Army circumference formula using neck and waist measurements. See your category instantly.
Health & Fitness
Body Fat Calculator
Estimate body fat percentage with the US Navy method using tape measurements. Track your fitness progress accurately for men and women.