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Quadratic Equation Solver
Solve quadratic equations in the form ax² + bx + c = 0
Enter Coefficients
Solutions
Why the Quadratic Formula Always Works
Factoring is faster when it works. But when you get something like x² + 3x + 1 = 0, factoring is impossible. The quadratic formula solves every quadratic equation, no matter what.
The formula: x = (-b ± √(b² - 4ac)) / 2a. Looks scary, but it's just plug-and-play. You get two solutions because of the ± symbol.
💡 Real Talk from Dr. Alex M., Ph.D.
Most students try to factor everything. Don't. If you can't see factors in 10 seconds, just use the formula. It's 100% reliable. I've seen too many test mistakes from students forcing factoring when it doesn't work.
Understanding the Discriminant
The discriminant is b² - 4ac. It tells you what kind of roots you'll get before you even solve:
- Δ > 0: Two different real roots (parabola crosses x-axis twice)
- Δ = 0: One repeated root (parabola touches x-axis once)
- Δ < 0: Two complex roots (parabola doesn't touch x-axis)
Example: For x² - 5x + 6 = 0, discriminant = 25 - 24 = 1 (positive). So you get 2 real roots: x = 2 and x = 3.
⚠️ Common Mistake: Sign Errors
When calculating -b in the formula, if b is negative, don't make it double negative. Example: If b = -5, then -b = 5. Also, watch that 2a is in the denominator for BOTH terms, not just the square root part.
Quick Tips
- Always simplify fractions: If you get x = 10/4, reduce to x = 5/2 or 2.5
- Check your work: Plug your answer back in. Does it make the equation = 0?
- Vertex formula: x-coordinate of vertex is -b/2a (same as the first part of quadratic formula)
- Graphing connection: The roots are where the graph crosses the x-axis
Example: x² - 4x + 4 = 0. Discriminant = 16 - 16 = 0, so one root. Using formula: x = (4 ± 0)/2 = 2. Double-checking: 4 - 8 + 4 = 0. Correct!
Reviewed by Dr. Alex M., Ph.D.
Mathematics Professor
Last updated: November 2025
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a. That's it. The ± means you get two answers (roots). Plug in your a, b, c values from ax² + bx + c = 0 and you're done. This is one of the most useful formulas in algebra.
What does the discriminant tell you?
The discriminant is b² - 4ac. If it's positive, you get 2 real roots. If it's zero, you get 1 repeated root. If it's negative, you get 2 complex (imaginary) roots. Basically, it tells you what kind of solution to expect before you even solve it.
Can every quadratic equation be solved?
Yes, as long as a ≠ 0. If a = 0, it's not a quadratic equation anymore - it's just bx + c = 0 (linear). When a ≠ 0, the quadratic formula works 100% of the time, even when factoring doesn't.
What are complex roots?
When discriminant < 0, you get imaginary numbers (with 'i' ). Example: x=2 + 3i and x=2 - 3i. They come in conjugate pairs. In real-life graphing, this means the parabola doesn't cross the x-axis at all.
How do I check if my answer is right?
Plug your root back into the original equation. If ax² + bx + c = 0, you should get 0 (or very close to 0 if you rounded). Example: For x² - 5x + 6 = 0, if x = 2, then 4 - 10 + 6 = 0. Works!