Quadratic Formula
ax² + bx + c = 0
Introduction:
Quadratic Formula Calculator is a very useful mathematical calculator that finds automatically the roots of any second degree polynomial equation. The user enters the coefficients (a, b, and c) and the calculator provides the users the roots instantly and step-by-step solution in complete format.The Quadratic Formula Calculator is a tool designed to make one of the most frequently-used calculations in algebra easy and convenient for both students and teachers. It will remove the arithmetical errors, save time and will make the user understand how the answer has been attained, rather than just what answer.All three types of roots two different real roots, one real repeated root and two complex (imaginary) roots are supported by our calculator. It displays the complete working for each problem, from finding the values of the coefficients, to calculating the discriminant, to presenting the roots the rigor expected in a school or professional environment.
What is a Quadratic Equation?
The word “quadratic” comes from the Latin word quadratus, meaning by “square.”
A quadratic equation is defined as:
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A polynomial expression which has the maximum degree of variable as 2. |
The standard form of a quadratic equation is:
ax² + bx + c = 0
where a, b, and c are real-number coefficients with a ≠ 0, and x is the unknown variable. The requirement that a ≠ 0 ensures the equation is genuinely quadratic (second-degree) rather than linear.
Quadratic Equation Formula
To find the values of x (called the roots or solutions) of a quadratic equation, we apply the Quadratic Formula:
This formula was derived by completing the square on the general form ax² + bx + c = 0. It always works for any quadratic equation with real or complex coefficients.
The expression inside the square root, b² − 4ac, is called the Discriminant (denoted Δ). It is the key that determines the nature of the roots before the full formula is applied.
Nature of the Roots
The discriminant Δ = b² − 4ac reveals the type of roots the equation will have:
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Discriminant (Δ) |
Condition |
Nature of Roots |
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Δ = b² − 4ac > 0 |
Positive |
Two distinct real roots |
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Δ = b² − 4ac = 0 |
Zero |
One repeated real root |
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Δ = b² − 4ac < 0 |
Negative |
Two complex (imaginary) roots |
How to Solve Quadratic Equations?
Follow these steps to solve any quadratic equation using the quadratic formula:
- Write the equation in standard form: ax² + bx + c = 0
- Identify the coefficients a, b, and c.
- Calculate the discriminant: Δ = b² − 4ac.
- Substitute into the formula and compute x₁ and x₂.
- Simplify and state the roots (real or complex).
Worked Example
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Solve: 6x² + 4x + 2 = 0 Step 1 — Identify coefficients: a = 6, b = 4, c = 2 Step 2 — Calculate the discriminant: Δ = b² − 4ac = (4)² − 4(6)(2) = 16 − 48 = −32 Since Δ < 0, the roots are complex (imaginary). Step 3 — Apply the quadratic formula: x = (−4 ± √(−32)) / (2 × 6) x = (−4 ± √(16 ×−2)) / 12 x = (−4 ± 4√2 · i) / 12 Step 4 — Simplify: x₁ = −1/3 + (√2/3)i ≈ −0.333 + 0.4714i x₂ = −1/3 − (√2/3)i ≈ −0.333 − 0.4714i |
Why Use This Calculator?
The Quadratic Formula Calculator has been designed to be accurate, fast and easy to use. Each of the results is accompanied by an all-in-one, step-by-step solution that users can follow and learn from. It deals with all three types of roots; allows for decimal and negative coefficients; and checks for errors in input.
Be it homework checking, test preparation or solving an engineering problem, this tool will give you instant and reliable results that you can trust and like our site.