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Quadratic Formula Calculator
For equation: ax² + bx + c = 0
Note: Δ (Delta) = b² - 4ac is called the discriminant
Solution Steps
Click Calculate to see step-by-step solution
Understanding the Quadratic Formula
The Quadratic Formula
Breaking Down the Formula
x₂ = (-b - √Δ) / 2a
Discriminant (Δ) Interpretation
- Δ > 0: Two distinct real solutions (the parabola crosses x-axis at two points)
- Δ = 0: One repeated real solution (the parabola touches x-axis at one point)
- Δ < 0: No real solutions (the parabola doesn't cross x-axis). Complex solutions exist.
Formula Components Explained
What each part means:
The "-b" Part
The negative of coefficient b. This shifts the solutions left or right depending on the sign of b.
The "±" Symbol
This means we perform two calculations: one with "+" and one with "-". This gives us two solutions (when they exist).
The "√(b² - 4ac)" Part
The square root of the discriminant. This represents the "spread" between the two solutions. When this is zero, both solutions are identical.
The "2a" Part
We divide by 2a (not just a). This ensures the formula accounts for the coefficient of x².
Example Calculation
Problem: x² - 5x + 6 = 0
Given: a = 1, b = -5, c = 6
Δ = 25 - 24
Δ = 1
x = [5 ± 1] / 2
x₂ = (5 - 1) / 2 = 4/2 = 2
Common Mistakes to Avoid
Mistake 1: Forgetting the ± Sign
Always use both + and -. You should get TWO solutions (unless Δ = 0). Using only one gives you only half the answer.
Mistake 2: Incorrect Sign of b
The formula uses "-b", not "b". If b = -5, then -b = 5. Be careful with signs!
Mistake 3: Not Simplifying the Square Root
Sometimes √Δ is not a perfect square. You may need to simplify it or leave it as a radical.
Mistake 4: Dividing by "a" Instead of "2a"
The denominator is always 2a, not just a. This is crucial for correct answers.
Mistake 5: Forgetting that a ≠ 0
If a = 0, you don't have a quadratic equation. The formula doesn't work. Check that a is not zero before starting!
Frequently Asked Questions
When was the quadratic formula discovered?
The quadratic formula was known to ancient Babylonians, but its modern form was developed by mathematicians over centuries.
Why is the ± symbol important?
The ± gives us two solutions because we can add or subtract √Δ. Parabolas typically cross the x-axis at two points.
What if the discriminant is negative?
Negative discriminant means no real solutions, only complex ones. The parabola doesn't cross the x-axis.
Can I use the formula for all equations?
The quadratic formula works for any equation in form ax² + bx + c = 0 where a ≠ 0. It's universal for quadratics!
Is there an easier way than the formula?
Factoring is sometimes easier if the equation factors nicely. But the formula works for ALL quadratic equations.
What if b² - 4ac = 0?
Then √Δ = 0, so both ± solutions give the same answer. You have one "double root" or "repeated root".
How do I know which root to use?
Both x₁ and x₂ are valid solutions. Use both unless the context specifies otherwise (e.g., physical constraint).
Can coefficients be decimals?
Yes! The formula works with any real numbers as coefficients. Decimals, fractions, negatives—all work.
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