Enter the coefficients for your rational equation in the form of:
(A*x + B) / (C*x + D) = (E*x + F) / (G*x + H)
Enter coefficients to find the solution(s).
Understanding and Solving Rational Equations
A rational equation is an equation that contains at least one rational expression. A rational expression is a fraction where the numerator and/or the denominator are polynomials. The general form of a rational equation we'll solve here is:
(A*x + B) / (C*x + D) = (E*x + F) / (G*x + H)
The goal is to find the value(s) of 'x' that satisfy this equality.
How the Solver Works:
The solver uses algebraic manipulation to transform the rational equation into a polynomial equation, which can then be solved. Here's the process:
Cross-Multiplication: To eliminate the denominators, we cross-multiply:
(A*x + B) * (G*x + H) = (E*x + F) * (C*x + D)
Expansion: Expand both sides of the equation using the distributive property (or FOIL method):
Let's define the coefficients for the resulting quadratic equation:
a = (A*G - E*C)
b = (A*H + B*G - E*D - F*C)
c = (B*H - F*D)
Solving the Quadratic Equation: Use the quadratic formula to find the values of x:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
The term (b^2 - 4ac) is the discriminant. Its value determines the nature of the solutions:
If discriminant > 0: Two distinct real solutions.
If discriminant = 0: One real solution (a repeated root).
If discriminant < 0: Two complex conjugate solutions (not handled by this basic solver, which focuses on real solutions).
Checking for Extraneous Solutions: A crucial step is to check if any of the calculated solutions make the original denominators zero. If C*x + D = 0 or G*x + H = 0 for a calculated value of x, that solution is extraneous and must be discarded. This solver checks for these conditions.
Since the discriminant is 0, there is one real solution:
x = [-b ± sqrt(0)] / (2a) = -2 / (2*1) = -1
Step 5 (Check for Extraneous Solutions):
Check if x = -1 makes denominators zero:
3x + 4 = 3*(-1) + 4 = -3 + 4 = 1 (Not zero)
7x + 8 = 7*(-1) + 8 = -7 + 8 = 1 (Not zero)
Since neither denominator is zero, x = -1 is a valid solution.
The solution is x = -1.
function solveRationalEquation() {
var a = parseFloat(document.getElementById("coeffA").value);
var b = parseFloat(document.getElementById("coeffB").value);
var c = parseFloat(document.getElementById("coeffC").value);
var d = parseFloat(document.getElementById("coeffD").value);
var e = parseFloat(document.getElementById("coeffE").value);
var f = parseFloat(document.getElementById("coeffF").value);
var g = parseFloat(document.getElementById("coeffG").value);
var h = parseFloat(document.getElementById("coeffH").value);
var resultDiv = document.getElementById("result");
var errorDiv = document.getElementById("error");
errorDiv.textContent = ""; // Clear previous errors
// Basic validation for coefficients being numbers
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d) || isNaN(e) || isNaN(f) || isNaN(g) || isNaN(h)) {
errorDiv.textContent = "Please enter valid numbers for all coefficients.";
resultDiv.innerHTML = "Enter coefficients to find the solution(s).";
return;
}
// Coefficients for the standard quadratic equation ax^2 + bx + c = 0
var quadA = (a * g) – (e * c);
var quadB = (a * h) + (b * g) – (e * d) – (f * c);
var quadC = (b * h) – (f * d);
// Handle the case where the equation simplifies to linear or is an identity/contradiction
if (quadA === 0) {
// It's a linear equation: quadB*x + quadC = 0
if (quadB === 0) {
if (quadC === 0) {
// 0 = 0, infinite solutions, but we need to check for extraneous roots
// For simplicity, we'll state it implies all real numbers are solutions
// unless they make original denominators zero.
// Determining specific excluded values for complex scenarios is hard.
// We'll focus on finding potential roots and checking them.
// If it reduces to 0=0, and no single 'x' value can be found,
// we state "All real numbers, except those that make denominators zero."
resultDiv.innerHTML = "The equation simplifies to an identity (0=0). All real numbers are potential solutions, provided they do not make the original denominators zero.";
} else {
// 0*x + C = 0, where C is non-zero, impossible.
errorDiv.textContent = "The equation has no solution.";
resultDiv.innerHTML = "No solution found.";
}
} else {
// Solve the linear equation: x = -quadC / quadB
var x = -quadC / quadB;
// Check for extraneous solutions
var denom1 = (c * x) + d;
var denom2 = (g * x) + h;
if (denom1 === 0 || denom2 === 0) {
errorDiv.textContent = "The only potential solution (x = " + x.toFixed(4) + ") is extraneous because it makes a denominator zero.";
resultDiv.innerHTML = "No valid solution found.";
} else {
resultDiv.innerHTML = "Unique solution: x = " + x.toFixed(4) + "";
}
}
} else {
// It's a quadratic equation: ax^2 + bx + c = 0
var discriminant = (quadB * quadB) – (4 * quadA * quadC);
if (discriminant < 0) {
errorDiv.textContent = "The equation has no real solutions (complex solutions exist).";
resultDiv.innerHTML = "No real solutions found.";
} else {
var x1 = (-quadB + Math.sqrt(discriminant)) / (2 * quadA);
var x2 = (-quadB – Math.sqrt(discriminant)) / (2 * quadA);
var solutions = [];
var extraneousFound = false;
// Check for extraneous solutions for x1
var denom1_x1 = (c * x1) + d;
var denom2_x1 = (g * x1) + h;
if (denom1_x1 !== 0 && denom2_x1 !== 0) {
solutions.push(x1.toFixed(4));
} else {
extraneousFound = true;
}
// Check for extraneous solutions for x2
// Avoid adding the same solution twice if discriminant is 0
if (discriminant !== 0) {
var denom1_x2 = (c * x2) + d;
var denom2_x2 = (g * x2) + h;
if (denom1_x2 !== 0 && denom2_x2 !== 0) {
solutions.push(x2.toFixed(4));
} else {
extraneousFound = true;
}
}
if (solutions.length === 0) {
errorDiv.textContent = "All potential solutions are extraneous.";
resultDiv.innerHTML = "No valid solution found.";
} else {
if (extraneousFound) {
errorDiv.textContent = "Note: One or more potential solutions were extraneous.";
}
resultDiv.innerHTML = "Solution(s): " + solutions.join(", ") + "";
}
}
}
}