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Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Standard Form
Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)
Key values: a = 1 · b = −5 · c = 6
Physics Problem
Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)
Key values: a = −4.9 · b = 20 · c = 1.5
Complex Roots
Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)
Key values: a = 1 · b = 2 · c = 5
This calculator is also known as Quadratic Formula Solver.
Read the complete guideThe Quadratic Formula Step by Step
To use the quadratic formula: (1) Write the equation in standard form ax² + bx + c = 0. (2) Identify a, b, and c. (3) Calculate the discriminant: b² - 4ac. (4) Substitute into x = (-b ± sqrt(discriminant)) / (2a). (5) Simplify to get both roots.
Examples
Complete Worked Example
Solve 2x² + 7x + 3 = 0 step by step.
Step 1: a=2, b=7, c=3. Step 2: Discriminant = 49 - 24 = 25. Step 3: sqrt(25) = 5. Step 4: x = (-7 ± 5) / 4. Step 5: x₁ = -2/4 = -0.5, x₂ = -12/4 = -3.
Key takeaway: The quadratic formula is a reliable algorithm: identify coefficients, compute discriminant, substitute, simplify.
Mastering the Quadratic Formula
Build fluency with the quadratic formula:
- Memorize the formula: x = (-b ± sqrt(b²-4ac)) / (2a)
- Always compute the discriminant first to know what kind of answer to expect
- Watch for sign errors: when b is negative, -b becomes positive
- The ± means you compute two values: one with + and one with -
- Simplify radicals when possible (e.g., sqrt(12) = 2*sqrt(3))
Frequently Asked Questions about Quadratic Formula Solver
Why is the quadratic formula important?
The quadratic formula is the only method that works for every quadratic equation, including those that cannot be factored. It also reveals the discriminant, which classifies the roots before you even solve.
How was the quadratic formula derived?
The formula is derived by completing the square on the general equation ax² + bx + c = 0. By systematically isolating x, you arrive at x = (-b ± sqrt(b²-4ac)) / (2a).
When should I use the quadratic formula instead of factoring?
Use the quadratic formula when the equation does not factor neatly (i.e., the roots are irrational or complex). Factoring is faster when the coefficients are small integers and the discriminant is a perfect square, but the formula always works.
Specialized Calculators
Choose from 5 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
Calculation Focus
2 CalculatorsSolving Method
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