Input your linear or quadratic equation to solve for 'x'.
Solving Steps:
Understanding and Using the Algebra Equation Solver
Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's used to express relationships between quantities and to solve for unknown values. At its core, algebra involves variables (like 'x'), coefficients, constants, and operations (addition, subtraction, multiplication, division).
Linear Equations
A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable. When graphed, a linear equation forms a straight line. The general form of a linear equation with one variable is typically written as ax + b = c, where 'x' is the variable we want to solve for, and 'a', 'b', and 'c' are constants.
How our solver handles linear equations:
For an equation like 2x + 5 = 17, the solver follows these steps:
Isolate the term with 'x': Subtract the constant term (5) from both sides. 2x = 17 - 5. This simplifies to 2x = 12.
Solve for 'x': Divide both sides by the coefficient of 'x' (2). x = 12 / 2. This gives the solution x = 6.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in one variable. The general form is ax² + bx + c = 0, where 'x' is the variable, and 'a', 'b', and 'c' are constants, with 'a' not equal to zero. Quadratic equations can have zero, one, or two real solutions.
How our solver handles quadratic equations:
Our solver primarily uses the quadratic formula to find the solutions for 'x' in equations of the form ax² + bx + c = 0. The quadratic formula is:
x = [-b ± √(b² - 4ac)] / 2a
The term inside the square root, b² - 4ac, is called the discriminant. Its value determines the nature of the roots:
If b² - 4ac > 0, there are two distinct real solutions.
If b² - 4ac = 0, there is exactly one real solution (a repeated root).
If b² - 4ac < 0, there are no real solutions (two complex conjugate solutions).
Since the discriminant is positive (4), there are two real solutions.
Apply the quadratic formula:
x = [ -(-4) ± √4 ] / 2(1)
x = [ 4 ± 2 ] / 2
Calculate the two solutions:
x1 = (4 + 2) / 2 = 6 / 2 = 3
x2 = (4 - 2) / 2 = 2 / 2 = 1
The solutions are x = 1 and x = 3.
Why Use an Algebra Solver?
Understanding how to solve equations manually is crucial for building strong mathematical foundations. However, an online solver provides several benefits:
Verification: Check your manual work and ensure accuracy.
Learning Tool: See the steps involved, helping you understand the process.
Efficiency: Quickly find solutions for complex problems, especially when time is limited.
Accessibility: Available anytime, anywhere, without the need for specialized software.
Our calculator is designed to be intuitive and provide clear, step-by-step guidance for solving common types of algebraic equations, making learning and problem-solving more accessible.
function solveEquation() {
var equationString = document.getElementById("equationInput").value.trim();
var resultDiv = document.getElementById("result");
var stepsDiv = document.getElementById("steps");
var stepsList = document.getElementById("stepsList");
// Clear previous results and steps
resultDiv.style.display = 'none';
stepsDiv.style.display = 'none';
stepsList.innerHTML = ";
if (!equationString) {
alert("Please enter an equation.");
return;
}
// Basic parsing and solving logic
// This is a simplified parser for common linear and quadratic forms.
// More complex equations would require a robust expression parser.
var cleanedEquation = equationString.toLowerCase().replace(/\s+/g, "); // Remove spaces
var linearMatch = cleanedEquation.match(/^([+-]?\d*\.?\d*)x([+-]\d+\.?\d+)?=([+-]?\d+\.?\d+)$/);
var quadraticMatch = cleanedEquation.match(/^([+-]?\d*\.?\d*)x\^2([+-]\d*\.?\d*)x([+-]\d+\.?\d+)?=0$/);
if (linearMatch) {
// Linear Equation: ax + b = c or ax = c or x + b = c etc.
// Capture groups:
// 1: Coefficient of x (a)
// 2: Constant term added/subtracted (b)
// 3: Right-hand side constant (c)
var aStr = linearMatch[1];
var bStr = linearMatch[2];
var cStr = linearMatch[3];
var a = parseFloat(aStr === " || aStr === '+' ? '1' : (aStr === '-' ? '-1' : aStr));
var b = bStr ? parseFloat(bStr) : 0;
var c = parseFloat(cStr);
// Ensure 'a' is not zero for a valid linear equation
if (isNaN(a) || isNaN(c)) {
alert("Invalid linear equation format. Please check your input.");
return;
}
if (a === 0) {
alert("Coefficient of 'x' cannot be zero for a linear equation.");
return;
}
var steps = [];
steps.push("Original Equation: " + equationString);
steps.push("Rearranging to isolate the 'x' term: " + a + "x = " + c + (b < 0 ? " – " : " + ") + Math.abs(b));
var newC = c – b;
steps.push("Simplifying the right side: " + a + "x = " + newC);
steps.push("Solving for x by dividing by the coefficient " + a + ": x = " + newC + " / " + a);
var x = newC / a;
steps.push("Solution: x = " + x);
displayResult(x, steps);
} else if (quadraticMatch) {
// Quadratic Equation: ax^2 + bx + c = 0
// Capture groups:
// 1: Coefficient of x^2 (a)
// 2: Coefficient of x (b)
// 3: Constant term (c)
var aStr = quadraticMatch[1];
var bStr = quadraticMatch[2];
var cStr = quadraticMatch[3];
var a = parseFloat(aStr === '' || aStr === '+' ? '1' : (aStr === '-' ? '-1' : aStr));
var b = parseFloat(bStr === '' || bStr === '+' ? '1' : (bStr === '-' ? '-1' : bStr));
var c = parseFloat(cStr);
// Handle cases like x^2 + 5 = 0 or 3x^2 – 12 = 0 where 'b' might be missing
// The regex captures '+bx' or '-bx', so if bStr is undefined, it means b=0
if (bStr === undefined) {
b = 0;
} else {
b = parseFloat(bStr === '' || bStr === '+' ? '1' : (bStr === '-' ? '-1' : bStr));
}
if (cStr === undefined) { // Case where c might be missing if the equation was e.g., x^2 + 2x = 0
c = 0;
} else {
c = parseFloat(cStr);
}
if (isNaN(a) || isNaN(b) || isNaN(c)) {
alert("Invalid quadratic equation format. Please check your input.");
return;
}
if (a === 0) {
alert("Coefficient 'a' cannot be zero for a quadratic equation. It becomes a linear equation.");
return;
}
var steps = [];
steps.push("Original Equation: " + equationString);
steps.push("Standard form identified: ax² + bx + c = 0");
steps.push("Coefficients: a = " + a + ", b = " + b + ", c = " + c);
var discriminant = b * b – 4 * a * c;
steps.push("Calculating the discriminant (Δ): b² – 4ac = " + b + "² – 4(" + a + ")(" + c + ") = " + discriminant);
if (discriminant < 0) {
steps.push("The discriminant is negative, indicating no real solutions (two complex solutions).");
resultDiv.innerHTML = "No real solutions";
resultDiv.style.display = 'block';
} else {
var sqrtDiscriminant = Math.sqrt(discriminant);
var x1 = (-b + sqrtDiscriminant) / (2 * a);
var x2 = (-b – sqrtDiscriminant) / (2 * a);
steps.push("Using the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a");
steps.push("Substituting values: x = [" + (-b) + " ± √" + discriminant + "] / " + (2 * a));
if (discriminant === 0) {
steps.push("The discriminant is zero, indicating one real solution (a repeated root).");
steps.push("Calculating the single solution: x = " + x1);
displayResult(x1, steps);
} else {
steps.push("Calculating the two distinct real solutions:");
steps.push("x1 = (" + (-b) + " + " + sqrtDiscriminant + ") / " + (2 * a) + " = " + x1);
steps.push("x2 = (" + (-b) + " – " + sqrtDiscriminant + ") / " + (2 * a) + " = " + x2);
resultDiv.innerHTML = "Solutions: x = " + x1.toFixed(4) + ", x = " + x2.toFixed(4);
resultDiv.style.display = 'block';
}
}
stepsDiv.style.display = 'block';
} else {
alert("Unsupported equation format. Please enter a linear equation (e.g., 2x + 3 = 9) or a quadratic equation in the form ax^2 + bx + c = 0 (e.g., x^2 – 5x + 6 = 0).");
}
}
function displayResult(solution, steps) {
var resultDiv = document.getElementById("result");
var stepsDiv = document.getElementById("steps");
var stepsList = document.getElementById("stepsList");
resultDiv.innerHTML = "Solution: x = " + solution.toFixed(4); // Display solution rounded to 4 decimal places
resultDiv.style.display = 'block';
for (var i = 0; i < steps.length; i++) {
var listItem = document.createElement("li");
listItem.textContent = steps[i];
stepsList.appendChild(listItem);
}
stepsDiv.style.display = 'block';
}