Quadratic Equation
GeneralQuadratic Equation
An equation of the form ax² + bx + c = 0, where a ≠ 0, characterized by a squared term. Solved using factoring, completing the square, or the quadratic formula.
Definition
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The squared term is what defines it as quadratic, remove it and the equation becomes linear instead, with a fundamentally different solution structure.
Quadratic equations appear across a surprising range of real-world contexts: the arc of a thrown object, the shape of a satellite dish or parabolic reflector, compound interest calculated over two periods, and problems involving area given fixed dimensions. The Quadratic Formula Calculator solves any quadratic equation directly once you know its a, b, and c coefficients.
Formula
Standard Form: ax² + bx + c = 0
Quadratic Formula: x = (−b ± √(b² − 4ac)) / 2a
The expression under the square root, b² − 4ac, is the discriminant, and its sign determines how many real solutions exist.
Worked Example
Solve 3x² − 5x − 2 = 0, where a = 3, b = −5, c = −2:
- Discriminant: (−5)² − 4(3)(−2) = 25 + 24 = 49
- x = (5 ± √49) / 6 = (5 ± 7) / 6
- x = 2 or x = −1/3
Both values, when substituted back into the original equation, satisfy it exactly, confirming the solutions.
Key Things to Know
- The coefficient a can never be zero. If a = 0, the squared term disappears and the equation becomes linear, not quadratic, by definition.
- Three solving methods exist for different situations. Factoring is fastest when it works cleanly, completing the square builds understanding and handles vertex form, and the quadratic formula solves any quadratic reliably regardless of how messy the numbers are.
- The discriminant predicts the solution count before you fully solve. Checking b² − 4ac first tells you whether to expect two, one, or zero real solutions.
- Graphically, solutions correspond to where the parabola crosses the x-axis. Two real solutions mean two crossing points, one means the vertex just touches the axis, none means the parabola never reaches it.
- Real-world applications span physics, engineering, and finance. Projectile motion, satellite dish geometry, and certain compounding interest problems all reduce to solving a quadratic equation.
Frequently Asked Questions