Quadratic Formula Calculator Formula
Understand the math behind the quadratic formula calculator. Each variable explained with a worked example.
Formulas Used
Root1
root1 = (-b + sqrt(abs(discriminant))) / (2 * a)Root2
root2 = (-b - sqrt(abs(discriminant))) / (2 * a)Discriminant Val
discriminant_val = discriminantVertex X
vertex_x = -b / (2 * a)Vertex Y
vertex_y = a * (-b / (2 * a))^2 + b * (-b / (2 * a)) + cVariables
| Variable | Description | Default |
|---|---|---|
a | Coefficient a | 1 |
b | Coefficient b | -5 |
c | Coefficient c | 6 |
discriminant | Derived value= b^2 - 4 * a * c | calculated |
How It Works
How to Use the Quadratic Formula
Formula
x = (-b ± √(b² - 4ac)) / (2a)
For the equation ax² + bx + c = 0
Discriminant (Δ = b² - 4ac)
Vertex
The vertex of the parabola is at x = -b/(2a), y = f(-b/(2a)).
Worked Example
Solve x² - 5x + 6 = 0 (a=1, b=-5, c=6).
- 01Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
- 02Since Δ > 0, there are two distinct real roots
- 03x₁ = (5 + √1) / 2 = 6 / 2 = 3
- 04x₂ = (5 - √1) / 2 = 4 / 2 = 2
- 05Vertex: x = 5/2 = 2.5, y = (2.5)² - 5(2.5) + 6 = -0.25
When to Use This Formula
- Solving any equation of the form ax² + bx + c = 0 — projectile motion, profit optimization, area problems, and any situation that produces a quadratic relationship.
- Finding the x-intercepts (roots) of a parabola for graphing, which tells you where a thrown object hits the ground or where revenue equals cost.
- Determining break-even points in business when the cost or revenue function is quadratic — the roots represent the quantities where profit switches between positive and negative.
- Analyzing the discriminant (b² - 4ac) alone to quickly determine whether a problem has two real solutions, one repeated solution, or no real solutions without solving fully.
- Solving quadratic equations that cannot be easily factored — the quadratic formula works for all quadratics, including those with irrational or complex roots.
Common Mistakes to Avoid
- Getting the sign wrong on b — the formula is x = (-b ± √(b²-4ac)) / 2a, with a negative sign on b. If the original equation has a negative coefficient for x (e.g., x² - 5x + 6), then b = -5 and -b = +5. Double-negatives here cause frequent sign errors.
- Dividing only part of the numerator by 2a — the entire expression (-b ± √(b²-4ac)) must be divided by 2a, not just the square root term. Writing x = -b ± √(b²-4ac)/2a is wrong because order of operations divides only the radical by 2a.
- Forgetting to rearrange the equation into standard form first — the formula requires ax² + bx + c = 0 with everything on one side. If the equation is x² + 3x = 10, you must rewrite it as x² + 3x - 10 = 0 before identifying a, b, and c.
- Ignoring the ± and reporting only one solution — the ± means there are typically two solutions (one with + and one with -). Reporting only one misses a valid root, which can mean overlooking a physically meaningful answer.
- Panicking when the discriminant is negative — b² - 4ac < 0 simply means no real roots (the parabola doesn't cross the x-axis). In applied contexts this means "no solution exists," which is itself a useful answer.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula x = (-b ± √(b²-4ac)) / (2a) finds the solutions (roots) of any quadratic equation ax² + bx + c = 0.
What does the discriminant tell us?
The discriminant (b²-4ac) tells you the nature of the roots: positive = 2 real roots, zero = 1 repeated root, negative = no real roots (2 complex roots).
What if the discriminant is negative?
If the discriminant is negative, the equation has no real solutions. The roots are complex numbers. This calculator shows the real part when the discriminant is negative.
Learn More
Guide
How to Solve Quadratic Equations - Complete Guide
Learn how to solve quadratic equations using factoring, the quadratic formula, and completing the square. Step-by-step methods with worked examples.
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