Solving Quadratic Equations Calculator

Quadratic Equation Solver

Enter the coefficients for the quadratic equation in the form ax² + bx + c = 0.







function calculateQuadratic() { var a = parseFloat(document.getElementById("coeffA").value); var b = parseFloat(document.getElementById("coeffB").value); var c = parseFloat(document.getElementById("coeffC").value); var resultDiv = document.getElementById("quadraticResult"); resultDiv.innerHTML = ""; // Clear previous results if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { if (b === 0) { if (c === 0) { resultDiv.innerHTML = "This is a trivial equation (0 = 0). It has infinite solutions."; } else { resultDiv.innerHTML = "This is an inconsistent equation (" + c + " = 0). It has no solution."; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation (a = 0). The solution is x = " + x.toFixed(4) + ""; } return; } var discriminant = b * b – 4 * a * c; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultDiv.innerHTML = "The equation has two distinct real roots:" + "x₁ = " + x1.toFixed(4) + "" + "x₂ = " + x2.toFixed(4) + ""; } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML = "The equation has one real root (or two identical real roots):" + "x = " + x.toFixed(4) + ""; } else { // discriminant < 0 var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = "The equation has two complex roots:" + "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i" + "x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding Quadratic Equations and Their Solutions

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • x represents the unknown variable.
  • a, b, and c are coefficients, with a not equal to zero.

The solutions to a quadratic equation are also known as its roots. These roots represent the x-intercepts of the parabola that the quadratic equation describes when graphed.

The Quadratic Formula

The most common method for solving quadratic equations is using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula allows you to find the values of x directly, given the coefficients a, b, and c.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, (b² - 4ac), which is called the discriminant. The value of the discriminant determines the nature of the roots:

  • If (b² - 4ac) > 0: The equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If (b² - 4ac) = 0: The equation has exactly one real root (sometimes called a repeated or double root). This means the parabola touches the x-axis at exactly one point (its vertex).
  • If (b² - 4ac) < 0: The equation has two complex (non-real) roots. These roots are conjugates of each other. In this case, the parabola does not intersect the x-axis.

How to Use the Quadratic Equation Solver

Our Quadratic Equation Solver simplifies the process of finding the roots of any quadratic equation. Simply follow these steps:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember, if a term is missing, its coefficient is 0 (e.g., in x² + 5 = 0, b = 0).
  2. Enter Values: Input the numerical values for Coefficient a, Coefficient b, and Coefficient c into the respective fields in the calculator.
  3. Solve: Click the "Solve Equation" button.
  4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex.

Examples of Quadratic Equations

Let's look at some examples to illustrate the different types of solutions:

Example 1: Two Distinct Real Roots

Consider the equation: x² - 3x + 2 = 0

  • a = 1
  • b = -3
  • c = 2

Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1 (which is > 0)

Using the calculator with these values will yield: x₁ = 2.0000, x₂ = 1.0000

Example 2: One Real Root

Consider the equation: x² - 4x + 4 = 0

  • a = 1
  • b = -4
  • c = 4

Discriminant = (-4)² – 4(1)(4) = 16 – 16 = 0

Using the calculator with these values will yield: x = 2.0000

Example 3: Two Complex Roots

Consider the equation: x² + 2x + 5 = 0

  • a = 1
  • b = 2
  • c = 5

Discriminant = (2)² – 4(1)(5) = 4 – 20 = -16 (which is < 0)

Using the calculator with these values will yield: x₁ = -1.0000 + 2.0000i, x₂ = -1.0000 – 2.0000i

Example 4: Linear Equation (a = 0)

Consider the equation: 0x² + 5x - 10 = 0 (or simply 5x - 10 = 0)

  • a = 0
  • b = 5
  • c = -10

Using the calculator with these values will yield: x = 2.0000

This calculator is a powerful tool for students, engineers, and anyone needing to quickly and accurately solve quadratic equations without manual calculation errors.

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