Quadratic Equation Solver
Solve quadratic equations of the form ax² + bx + c = 0. Get real and complex roots, step-by-step solutions, and a visual graph of the parabola. Perfect for students, teachers, and math enthusiasts.
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation of the form:
ax² + bx + c = 0
Where a, b, and c are constants, and a ≠ 0.
Quadratic equations are fundamental in algebra and appear in many real-world applications, including physics (projectile motion), engineering, economics, and optimization problems. The graph of a quadratic equation is a parabola that opens upward (if a > 0) or downward (if a < 0).
How Does the Quadratic Equation Solver Work?
This solver uses the quadratic formula to find the roots of any quadratic equation:
x = (−b ± √Δ) / (2a)
Where Δ = b² − 4ac is the discriminant.
The discriminant determines the nature of the roots:
- Δ > 0: Two distinct real roots.
- Δ = 0: One real root (repeated).
- Δ < 0: Two complex conjugate roots.
The calculator also displays:
- Sum of Roots: α + β = −b/a
- Product of Roots: α × β = c/a
- Vertex: The minimum or maximum point of the parabola.
- Axis of Symmetry: x = −b/(2a)
❓ Quadratic Equation Solver FAQ
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic equation of the form ax² + bx + c = 0.
What is the discriminant?
The discriminant is Δ = b² − 4ac. It determines the nature of the roots: if Δ > 0, there are two real roots; if Δ = 0, there is one real root; if Δ < 0, there are two complex roots.
What does it mean if the discriminant is negative?
If the discriminant is negative, the quadratic equation has no real roots. Instead, it has two complex conjugate roots, which involve the imaginary unit i = √(−1).
What is the sum of roots formula?
For a quadratic equation ax² + bx + c = 0, the sum of the roots is α + β = −b/a. This is known as Vieta's formula.
What is the product of roots formula?
For a quadratic equation ax² + bx + c = 0, the product of the roots is α × β = c/a. This is also a Vieta's formula.
What is the vertex of a parabola?
The vertex is the minimum or maximum point of the parabola. Its x-coordinate is x = −b/(2a), and the y-coordinate is found by substituting this x-value into the equation.
What is the axis of symmetry?
The axis of symmetry is the vertical line that passes through the vertex of the parabola. Its equation is x = −b/(2a).
How do I solve a quadratic equation by factoring?
To solve by factoring, rewrite the equation as (x − p)(x − q) = 0, where p and q are the roots. Then set each factor equal to zero. This method works when the equation has integer roots.
What is completing the square?
Completing the square is a method of solving quadratic equations by rewriting the equation in the form (x − h)² = k. It's useful for deriving the quadratic formula and finding the vertex of a parabola.
What is the difference between real and complex roots?
Real roots are numbers that exist on the number line. Complex roots involve the imaginary unit i and are of the form a ± bi. Complex roots occur when the discriminant is negative.
What is the significance of the coefficient 'a'?
The coefficient 'a' determines the direction and width of the parabola. If a > 0, the parabola opens upward (minimum). If a < 0, the parabola opens downward (maximum). The larger |a|, the narrower the parabola.
What is a quadratic equation used for in real life?
Quadratic equations are used in physics (projectile motion, free fall), engineering (bridge design, structural analysis), economics (profit maximization), and computer graphics (parabolic curves).
What is the difference between a quadratic equation and a quadratic function?
A quadratic equation is set equal to zero (ax² + bx + c = 0) and is solved for x. A quadratic function is f(x) = ax² + bx + c and represents a parabola when graphed. The roots of the equation are the x-intercepts of the function.
How do I check my quadratic equation solutions?
Substitute each root back into the original equation. The equation should equal zero for both roots. This calculator shows the roots and you can verify them manually.
What is the quadratic formula derived from?
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. It provides a direct method for finding the roots of any quadratic equation.
Can I use this calculator for equations with fractions?
Yes. Enter the coefficients as decimals or fractions. For example, you can enter 0.5 for 1/2. The calculator will handle decimal coefficients accurately.
What is the difference between roots and solutions?
In the context of quadratic equations, "roots" and "solutions" mean the same thing — the values of x that satisfy the equation. Both terms are used interchangeably.
How accurate are the results from this calculator?
This calculator uses JavaScript's floating-point arithmetic and provides results with up to 5 decimal places of precision. For most practical purposes, this is sufficient. For exact results, you may need to use symbolic calculation tools.