Step by Step Quadratic Equation Solver
Solve quadratic equations step by step. Get detailed solutions, factorization, and graph analysis. Free calculator.
Quick Answer
Quadratic equations ax² + bx + c = 0 have solutions using the quadratic formula. The discriminant (b² - 4ac) determines the nature of roots: positive (2 real roots), zero (1 repeated root), or negative (2 complex roots).
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Calculate With Full ToolWhat is Step by Step Quadratic Equation Solver?
Step by step quadratic equation solver provides detailed solutions for equations in the form ax² + bx + c = 0. This method breaks down the quadratic formula into understandable steps, showing how to calculate discriminant, apply the quadratic formula, and interpret results.
How Step-by-Step Works
The solver follows a systematic approach: identify coefficients, calculate discriminant, determine root type, apply appropriate formula, and verify solutions. Each step is explained with mathematical reasoning and practical examples for better understanding.
Why Step-by-Step Matters
Step-by-step solutions help students understand the underlying mathematics, identify common mistakes, and develop problem-solving skills. This approach builds confidence in algebraic manipulation and equation solving techniques.
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
Root Types: Δ > 0 (2 real), Δ = 0 (1 repeated), Δ < 0 (2 complex)
Vertex: x = -b/2a, y = -Δ/4a
Axis of Symmetry: x = -b/2a
Step-by-Step Example
Example: x² - 5x + 6 = 0
Step 1: Identify coefficients: a = 1, b = -5, c = 6
Step 2: Calculate discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
Step 3: Since Δ > 0, two real roots exist
Step 4: Apply quadratic formula: x = (5 ± √1) / 2
Step 5: Calculate roots: x₁ = (5 + 1) / 2 = 3, x₂ = (5 - 1) / 2 = 2
Step 6: Verify: 3² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓
This example shows how the step-by-step method breaks down solving x² - 5x + 6 = 0 into clear, logical steps. The discriminant calculation reveals two real roots, and the quadratic formula provides the exact solutions x = 3 and x = 2.
Who Should Use This Calculator?
Students
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Teachers
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Engineers
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Parents
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Frequently Asked Questions
What is the quadratic formula?
The quadratic formula x = (-b ± √(b² - 4ac)) / 2a solves equations ax² + bx + c = 0. It provides exact solutions for any quadratic equation, with the discriminant determining the nature of roots (real, repeated, or complex).
How do you find the discriminant?
The discriminant Δ = b² - 4ac determines root nature. Δ > 0: two distinct real roots, Δ = 0: one repeated real root, Δ < 0: two complex conjugate roots. This is the key to understanding quadratic equation behavior.
What if a = 0?
If a = 0, the equation becomes linear: bx + c = 0. Solve for x = -c/b. This is a special case that doesn't require the quadratic formula and has exactly one solution.
How do you check your answer?
Substitute each root back into the original equation. If the result equals zero, your solutions are correct. This verification step is crucial for avoiding calculation errors.
What are complex roots?
Complex roots occur when Δ < 0. They are expressed as a ± bi where i is the imaginary unit. These represent solutions in the complex number system and have important applications in engineering.
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