Quadratic Equation Calculator
Solve ax² + bx + c = 0 with complete step-by-step solutions
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0. The solutions to this equation are called roots and can be found using multiple methods including the quadratic formula, factoring, completing the square, or graphing.
The Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / 2a provides a systematic method to find the roots of any quadratic equation. The expression under the square root (b² - 4ac) is called the discriminant and determines the nature of the roots.
Discriminant Analysis
- If Δ > 0: Two distinct real roots
- If Δ = 0: One repeated real root (perfect square)
- If Δ < 0: Two complex conjugate roots
Methods of Solving
Quadratic Formula: Works for all quadratic equations. Most reliable method when factoring is difficult or impossible.
Factoring: Useful when the equation can be expressed as a product of binomials. Requires integer or simple rational roots.
Completing the Square: Converts the equation to vertex form and is useful for understanding parabola properties.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed.
Why must coefficient a be non-zero?
If a = 0, the equation becomes bx + c = 0, which is linear (first-degree), not quadratic. The x² term is essential for the equation to be quadratic.
What are complex roots?
Complex roots occur when the discriminant is negative. They come in conjugate pairs (a + bi and a - bi) where i is the imaginary unit (√-1). These roots are not real numbers.
How do I know which method to use?
Use factoring if you can easily identify factors. Use the quadratic formula for any equation, especially when factoring is difficult. Use completing the square when you need vertex form or for derivation purposes.
Can a quadratic equation have one solution?
Yes, when the discriminant equals zero, the equation has one repeated root (a double root). This occurs when the parabola touches the x-axis at exactly one point (the vertex).
What is the relationship between roots and factors?
If r₁ and r₂ are roots, the equation can be factored as a(x - r₁)(x - r₂) = 0. This shows that the roots are the values that make each factor equal to zero.
How accurate is the quadratic formula?
The quadratic formula is mathematically exact. Any rounding errors come from decimal approximations in computation. For exact answers, keep results in radical form when possible.
What are real-world applications?
Quadratic equations model projectile motion, optimization problems, area calculations, profit analysis, and many physics phenomena where relationships involve squared terms.
Related Calculators
Quadratic Formula Calculator
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Completing the Square Calculator
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Parabola Calculator
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