Enter two algebraic expressions to determine if they are mathematically equivalent.
What are Equivalent Expressions?
In algebra, equivalent expressions are two or more algebraic statements that always result in the same numerical value, regardless of which number is substituted for the variable (usually 'x'). While they may look visually different, they represent the same mathematical relationship.
Common Algebraic Properties
Expressions are typically proven equivalent through the application of math rules:
Distributive Property: a(b + c) = ab + ac. For example, 3(x + 2) is equivalent to 3x + 6.
Commutative Property: a + b = b + a. For example, x + 5 is equivalent to 5 + x.
Associative Property: (a + b) + c = a + (b + c). For example, (x + 2) + 3 is equivalent to x + 5.
Combining Like Terms: 2x + 4x is equivalent to 6x.
Example Comparison
Expression 1
Expression 2
Equivalent?
5(x – 1)
5x – 5
Yes
x^2 + x
x(x + 1)
Yes
3x + 4
7x
No
How to Use the Calculator
Enter your first expression in the top field. Use 'x' as your variable.
Enter the second expression in the bottom field.
Use ^ for exponents (e.g., x^2).
Click Verify Equivalence. The tool tests various values of 'x' to see if the outputs match consistently.
function checkEquivalency() {
var raw1 = document.getElementById("expr1").value;
var raw2 = document.getElementById("expr2").value;
var resultDiv = document.getElementById("calcResult");
if (!raw1 || !raw2) {
resultDiv.style.display = "block";
resultDiv.style.backgroundColor = "#fdecea";
resultDiv.style.color = "#d32f2f";
resultDiv.innerHTML = "Please enter both expressions to compare.";
return;
}
function formatExpression(str) {
var s = str.toLowerCase();
// Replace ^ with ** for JS power
s = s.replace(/\^/g, "**");
// Insert * between number and x (e.g., 2x -> 2*x)
s = s.replace(/(\d)([a-z])/g, "$1*$2");
// Insert * between x and number (e.g., x2 -> x*2)
s = s.replace(/([a-z])(\d)/g, "$1*$2");
// Insert * between ) and ( (e.g., (x+1)(x+2))
s = s.replace(/\)\(/g, ")*(");
// Insert * between number and (
s = s.replace(/(\d)\(/g, "$1*(");
// Insert * between ) and number
s = s.replace(/\)(\d)/g, ")*$1");
// Insert * between x and (
s = s.replace(/([a-z])\(/g, "$1*(");
// Insert * between ) and x
s = s.replace(/\)([a-z])/g, ")*$1");
return s;
}
var f1Str = formatExpression(raw1);
var f2Str = formatExpression(raw2);
var testValues = [-10, -5, -1, 0, 0.5, 1, 2, 5, 10, 100];
var isEquivalent = true;
var hasError = false;
try {
for (var i = 0; i 0.0000001) {
isEquivalent = false;
break;
}
}
} catch (e) {
hasError = true;
}
resultDiv.style.display = "block";
if (hasError) {
resultDiv.style.backgroundColor = "#fff4e5";
resultDiv.style.color = "#663c00";
resultDiv.innerHTML = "Error: Invalid syntax. Use standard algebraic format (e.g., 2(x+1) or x^2).";
} else if (isEquivalent) {
resultDiv.style.backgroundColor = "#e8f5e9";
resultDiv.style.color = "#2e7d32";
resultDiv.innerHTML = "✓ The expressions are Equivalent!";
} else {
resultDiv.style.backgroundColor = "#fdecea";
resultDiv.style.color = "#d32f2f";
resultDiv.innerHTML = "✗ The expressions are NOT Equivalent.";
}
}