Calculator for Precalculus

Quadratic Equation Solver (Precalculus)

Result:

Enter coefficients and click 'Calculate Roots' to see the solution.
function calculateQuadraticRoots() { var a = parseFloat(document.getElementById("coefficientA").value); var b = parseFloat(document.getElementById("coefficientB").value); var c = parseFloat(document.getElementById("coefficientC").value); var resultDiv = document.getElementById("result"); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { // Not a quadratic equation if (b === 0) { if (c === 0) { resultDiv.innerHTML = "This equation is 0 = 0, which has infinite solutions."; } else { resultDiv.innerHTML = "This equation is " + c + " = 0, which 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 root 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 (repeated root):x = " + x.toFixed(4); } else { // Discriminant < 0, complex roots var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = "The equation has two complex conjugate roots:x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding Quadratic Equations in Precalculus

Quadratic equations are a fundamental concept in precalculus, forming the basis for understanding parabolas, optimization problems, and more complex functions. 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, with a ≠ 0.
  • x represents the unknown variable.

The Quadratic Formula

The most common method for solving quadratic equations is using the quadratic formula. This formula provides the values of x (the roots or solutions) that satisfy the equation. The quadratic formula is:

x = [-b ± √(b² – 4ac)] / 2a

Our calculator above uses this formula to find the roots of any quadratic equation you input.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, (b² - 4ac), known as the discriminant. The value of the discriminant tells us about the nature of the roots:

  • If Discriminant > 0: The equation has two distinct real roots. This means the parabola (the graph of the quadratic function) intersects the x-axis at two different points.
  • If Discriminant = 0: The equation has exactly one real root (also called a repeated or double root). The parabola touches the x-axis at exactly one point (its vertex).
  • If Discriminant < 0: The equation has two complex conjugate roots. This means the parabola does not intersect the x-axis; its roots involve imaginary numbers.

Examples of Quadratic Equations and Their Solutions

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

Example 1: Two Distinct Real Roots

Consider the equation: x² - 5x + 6 = 0

  • a = 1, b = -5, c = 6
  • Discriminant = (-5)² – 4(1)(6) = 25 – 24 = 1
  • Since 1 > 0, there are two distinct real roots.
  • Using the formula: x = [5 ± √1] / 2(1)
  • x₁ = (5 + 1) / 2 = 3
  • x₂ = (5 – 1) / 2 = 2

(Try these values in the calculator: a=1, b=-5, c=6)

Example 2: One Real Root (Repeated)

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

  • a = 1, b = 4, c = 4
  • Discriminant = (4)² – 4(1)(4) = 16 – 16 = 0
  • Since 0 = 0, there is one real root.
  • Using the formula: x = [-4 ± √0] / 2(1)
  • x = -4 / 2 = -2

(Try these values in the calculator: a=1, b=4, c=4)

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + x + 1 = 0

  • a = 1, b = 1, c = 1
  • Discriminant = (1)² – 4(1)(1) = 1 – 4 = -3
  • Since -3 < 0, there are two complex conjugate roots.
  • Using the formula: x = [-1 ± √-3] / 2(1)
  • x = [-1 ± i√3] / 2
  • x₁ = -0.5 + 0.8660i
  • x₂ = -0.5 – 0.8660i

(Try these values in the calculator: a=1, b=1, c=1)

Special Case: When 'a' is Zero

If the coefficient 'a' is 0, the equation is no longer quadratic. It becomes a linear equation: bx + c = 0. In this case, if b ≠ 0, there is one real solution: x = -c/b. If both a and b are 0, the equation simplifies to c = 0, which either has infinite solutions (if c=0) or no solution (if c≠0).

Understanding quadratic equations and their solutions is a cornerstone of precalculus, preparing students for advanced topics in calculus and other STEM fields.

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