Quadratic Formula Calculator

Solve equations of the form ax² + bx + c = 0

Coefficients

Coefficient a (≠ 0)

Coefficient b

Coefficient c

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Precision (Decimal Places) 0 1 2 3 4 5 6 7 8

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Solution

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What Is the Quadratic Formula?

The quadratic formula is a fundamental theorem in algebra used for solving quadratic equations, which are polynomial equations of the second degree. The standard form of a quadratic equation is:

ax² + bx + c = 0

Here, x represents an unknown variable, while a, b, and c are known numbers, or coefficients. The coefficient a must not be zero for the equation to be quadratic. The formula provides the values of x (the roots) that satisfy the equation:

x = [-b ± √(b² - 4ac)] / 2a

This single formula elegantly yields the solutions, whether they are real or complex numbers.

How to Use the Quadratic Formula (Step-by-Step Example)

Let's solve the equation 2x² - 5x + 2 = 0 using the formula.

1. Identify Coefficients: First, identify a, b, and c. Here, a = 2, b = -5, and c = 2.
2. Calculate the Discriminant: The expression inside the square root, b² - 4ac, is called the discriminant (Δ). It tells us about the nature of the roots.
   Δ = (-5)² - 4(2)(2) = 25 - 16 = 9.
3. Apply the Formula: Substitute the coefficients and the discriminant into the quadratic formula.
   x = [-(-5) ± √9] / (2 * 2)
   x = [5 ± 3] / 4
4. Calculate the Two Roots: The '±' symbol means we have two solutions.
   Root 1 (x₁): (5 + 3) / 4 = 8 / 4 = 2.
   Root 2 (x₂): (5 - 3) / 4 = 2 / 4 = 0.5.

So, the solutions to 2x² - 5x + 2 = 0 are x = 2 and x = 0.5.

The Discriminant & What It Tells You

The discriminant (Δ = b² - 4ac) is a powerful part of the formula because it reveals the number and type of roots without fully solving the equation.

• If Δ > 0 (Positive): The equation has two distinct real roots. The parabola intersects the x-axis at two different points.
• If Δ = 0 (Zero): The equation has exactly one real root (also called a repeated or double root). The vertex of the parabola touches the x-axis at a single point.
• If Δ < 0 (Negative): The equation has no real roots. The square root of a negative number results in imaginary numbers, leading to two complex conjugate roots. The parabola does not intersect the x-axis at all.

Vertex, Axis of Symmetry & Graphing Parabolas

A quadratic equation graphically represents a U-shaped curve called a parabola.

• Orientation: If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
• Vertex: This is the highest or lowest point of the parabola. Its coordinates (h, k) can be found with:
  h = -b / 2a
k = f(h) = a(h)² + b(h) + c
• Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two mirror images. Its equation is x = h.

The roots, vertex, and y-intercept (the point where x=0, which is simply `c`) are key points for sketching the graph of the parabola.

Frequently Asked Questions

Q: What happens if 'a' is zero?
A: If a = 0, the equation is no longer quadratic; it becomes a linear equation (bx + c = 0). The solution is simply x = -c / b. Our calculator detects this and provides the linear solution.

Q: Can the quadratic formula be used for any polynomial?
A: No, it is specifically for second-degree polynomials (quadratics). Third-degree (cubic) and fourth-degree (quartic) equations have their own, much more complex formulas, and there is no general algebraic formula for polynomials of degree five or higher.

Q: What is a "numerically stable" formula?
A: When `b` is very large compared to `a` and `c`, the standard formula can suffer from a "catastrophic cancellation" error due to floating-point precision limits in computers. This can make one of the roots inaccurate. A numerically stable alternative formula is used in such cases to maintain precision.

This calculator provides mathematical solutions and educational explanations. Verify results for critical applications.