Mathpapa Algebra Calculator

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Quadratic Equation Solver

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

Results will appear here.

function calculateQuadratic() { var a = parseFloat(document.getElementById('coefficientA').value); var b = parseFloat(document.getElementById('coefficientB').value); var c = parseFloat(document.getElementById('constantC').value); var resultDiv = document.getElementById('quadraticResult'); var output = "; if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { // This is a linear equation: bx + c = 0 if (b === 0) { if (c === 0) { output = 'This equation simplifies to 0 = 0, which means there are infinite solutions.'; } else { output = 'This equation simplifies to ' + c + ' = 0, which is false. Therefore, there is no solution.'; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; output = 'Since coefficient \'a\' is 0, this is a linear equation.'; output += 'The solution is: x = ' + x.toFixed(4) + ''; } } else { // Standard quadratic equation: ax^2 + bx + c = 0 var discriminant = b * b – 4 * a * c; if (discriminant >= 0) { // Real roots var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); if (discriminant === 0) { output = 'The equation has one real (repeated) root:'; output += 'x = ' + x1.toFixed(4) + ''; } else { output = 'The equation has two distinct real roots:'; output += 'x₁ = ' + x1.toFixed(4) + ''; output += 'x₂ = ' + x2.toFixed(4) + ''; } } else { // Complex roots var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); output = 'The equation has two complex conjugate roots:'; output += 'x₁ = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i'; output += 'x₂ = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; } } resultDiv.innerHTML = output; }

Understanding and Solving Quadratic Equations

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It provides a powerful framework for solving problems across various disciplines, from science and engineering to finance and everyday decision-making. One of the most common and important types of equations you'll encounter in algebra is the quadratic equation.

What is a Quadratic Equation?

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

ax² + bx + c = 0

Where:

  • a, b, and c are coefficients (real numbers).
  • x represents the unknown variable.
  • The coefficient a cannot be equal to 0. If 'a' were 0, the x² term would vanish, and the equation would become a linear equation (bx + c = 0).

The solutions to a quadratic equation are also known as its "roots" or "zeros." These are the values of 'x' that make the equation true.

The Quadratic Formula

The most common and reliable method for finding the roots of any quadratic equation is by using the quadratic formula. This formula is derived by completing the square on the standard form of the quadratic equation:

x = [-b ± √(b² – 4ac)] / 2a

This formula provides two potential solutions for 'x', corresponding to the '+' and '-' signs before the square root.

The Discriminant (Δ)

A crucial part of the quadratic formula is the expression under the square root: b² – 4ac. This is called the discriminant (often denoted by Δ). The value of the discriminant tells us about the nature of the roots:

  • If Δ > 0: There are two distinct real roots. The parabola (graph of the quadratic equation) intersects the x-axis at two different points.
  • If Δ = 0: There is exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point (its vertex).
  • If Δ < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis.

Special Cases: When 'a' is Zero

As mentioned, a quadratic equation requires 'a' to be non-zero. However, our calculator handles cases where 'a' is entered as 0, effectively treating it as a linear equation or a constant equation:

  • If a = 0 and b ≠ 0: The equation becomes bx + c = 0, which is a simple linear equation. The solution is x = -c/b.
  • If a = 0 and b = 0:
    • If c = 0: The equation becomes 0 = 0, which is always true. This means there are infinite solutions.
    • If c ≠ 0: The equation becomes c = 0 (e.g., 5 = 0), which is never true. This means there is no solution.

How to Use the Quadratic Equation Solver

Our calculator simplifies the process of finding the roots of any quadratic equation. Follow these steps:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for 'a', 'b', and 'c'. Remember the standard form: ax² + bx + c = 0.
  2. Enter Values: Input the numerical values for 'a', 'b', and 'c' into the respective fields in the calculator above.
  3. Click Calculate: Press the "Calculate Roots" button.
  4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex, and if there's one or two distinct solutions. It also handles the special cases where 'a' is zero.

Examples of Quadratic Equations and Their Solutions

Example 1: Two Distinct Real Roots

Equation: x² - 5x + 6 = 0

  • a = 1
  • b = -5
  • c = 6

Calculation:
Discriminant (Δ) = (-5)² – 4(1)(6) = 25 – 24 = 1
Since Δ > 0, there are two distinct real roots.
x = [5 ± √1] / 2(1)
x₁ = (5 + 1) / 2 = 3
x₂ = (5 – 1) / 2 = 2

Calculator Input: a=1, b=-5, c=6
Calculator Output: x₁ = 3.0000, x₂ = 2.0000

Example 2: One Real (Repeated) Root

Equation: x² - 4x + 4 = 0

  • a = 1
  • b = -4
  • c = 4

Calculation:
Discriminant (Δ) = (-4)² – 4(1)(4) = 16 – 16 = 0
Since Δ = 0, there is one real (repeated) root.
x = [4 ± √0] / 2(1)
x = 4 / 2 = 2

Calculator Input: a=1, b=-4, c=4
Calculator Output: x = 2.0000

Example 3: Two Complex Conjugate Roots

Equation: x² + x + 1 = 0

  • a = 1
  • b = 1
  • c = 1

Calculation:
Discriminant (Δ) = (1)² – 4(1)(1) = 1 – 4 = -3
Since Δ < 0, there are two complex conjugate roots.
x = [-1 ± √-3] / 2(1)
x = [-1 ± i√3] / 2
x₁ = -0.5 + 0.8660i
x₂ = -0.5 – 0.8660i

Calculator Input: a=1, b=1, c=1
Calculator Output: x₁ = -0.5000 + 0.8660i, x₂ = -0.5000 – 0.8660i

Example 4: Linear Equation (a=0)

Equation: 0x² + 2x - 4 = 0 (effectively 2x - 4 = 0)

  • a = 0
  • b = 2
  • c = -4

Calculation:
2x = 4
x = 4 / 2 = 2

Calculator Input: a=0, b=2, c=-4
Calculator Output: Since coefficient 'a' is 0, this is a linear equation. The solution is: x = 2.0000

Example 5: No Solution (a=0, b=0, c≠0)

Equation: 0x² + 0x + 5 = 0 (effectively 5 = 0)

  • a = 0
  • b = 0
  • c = 5

Calculation:
The equation simplifies to 5 = 0, which is false.

Calculator Input: a=0, b=0, c=5
Calculator Output: This equation simplifies to 5 = 0, which is false. Therefore, there is no solution.

Mastering quadratic equations is a key step in developing strong algebraic skills. This calculator serves as a helpful tool to quickly find solutions and verify your manual calculations, enhancing your understanding of this fundamental mathematical concept.

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