Enter the coefficients 'a', 'b', and 'c' for your quadratic equation (ax² + bx + c = 0) to see the step-by-step factoring process.
Understanding Quadratic Factoring Step-by-Step
Factoring a quadratic equation is a fundamental skill in algebra, allowing us to find the roots (or solutions) of the equation ax² + bx + c = 0. This process essentially rewrites the quadratic expression into a product of two linear expressions. Our calculator breaks down this process, especially for trinomials of the form ax² + bx + c.
The Goal of Factoring
The primary goal is to express the quadratic expression ax² + bx + c as a product of two binomials, like (px + q)(rx + s). If we can achieve this, setting the expression to zero, (px + q)(rx + s) = 0, allows us to easily find the values of x for which the equation holds true by setting each binomial factor to zero: px + q = 0 and rx + s = 0.
Methods for Factoring
The method used often depends on the value of the leading coefficient 'a':
When a = 1: We look for two numbers that multiply to 'c' and add up to 'b'. If these numbers are 'm' and 'n', the factored form is (x + m)(x + n).
When a ≠ 1: This is more complex. Common methods include:
AC Method (Grouping): Find two numbers that multiply to a*c and add up to 'b'. Rewrite the middle term 'bx' using these two numbers, then factor by grouping.
Trial and Error: Guess binomial factors and expand them to see if they match the original trinomial.
Using the Quadratic Formula: While not strictly factoring, the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a gives the roots. If the roots are r₁ and r₂, the factored form is a(x - r₁)(x - r₂).
Our calculator primarily focuses on the process that leads to finding the factors, often using principles similar to the AC method or by finding the roots first and then constructing the factors.
How the Calculator Works (Conceptual Steps)
Input Coefficients: You provide the values for 'a', 'b', and 'c'.
Calculate Discriminant (if applicable): For real roots, the discriminant (Δ = b² – 4ac) must be non-negative. This is a key check.
Find Roots (using Quadratic Formula): The calculator uses the quadratic formula to find the roots, x₁ and x₂.
Construct Factors: Using the roots, the factored form is derived as a(x - x₁)(x - x₂). The calculator will simplify this into the form (px + q)(rx + s) where possible, or present the steps to reach it.
Handle Special Cases: It checks for cases where 'a' is 1, or where factoring might not result in simple integer coefficients.
Example: Factoring x² + 5x + 6
Let's factor the equation where a=1, b=5, and c=6.
Identify: a=1, b=5, c=6.
Find two numbers: We need two numbers that multiply to c (6) and add to b (5). These numbers are 2 and 3 (2 * 3 = 6 and 2 + 3 = 5).
Write factored form: Since a=1, the factored form is (x + 2)(x + 3).
This calculator provides a guided approach to understanding and performing these factoring steps.
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b) {
var t = b;
b = a % b;
a = t;
}
return a;
}
function simplifyFraction(numerator, denominator) {
if (denominator === 0) return { num: numerator, den: denominator, str: numerator + "/0 (Undefined)" };
if (numerator === 0) return { num: 0, den: 1, str: "0" };
var commonDivisor = gcd(numerator, denominator);
var simplifiedNumerator = numerator / commonDivisor;
var simplifiedDenominator = denominator / commonDivisor;
// Ensure denominator is positive
if (simplifiedDenominator < 0) {
simplifiedNumerator = -simplifiedNumerator;
simplifiedDenominator = -simplifiedDenominator;
}
var fractionString = simplifiedNumerator.toString();
if (simplifiedDenominator !== 1) {
fractionString += "/" + simplifiedDenominator;
}
return { num: simplifiedNumerator, den: simplifiedDenominator, str: fractionString };
}
function calculateFactoring() {
var stepsDiv = document.getElementById("calculationSteps");
var errorDiv = document.getElementById("errorMessage");
stepsDiv.innerHTML = "
Factoring Steps:
"; // Clear previous steps
errorDiv.textContent = ""; // Clear previous errors
var aStr = document.getElementById("quadraticA").value.trim();
var bStr = document.getElementById("quadraticB").value.trim();
var cStr = document.getElementById("quadraticC").value.trim();
if (aStr === "" || bStr === "" || cStr === "") {
errorDiv.textContent = "Please fill in all coefficient fields.";
return;
}
var a = parseFloat(aStr);
var b = parseFloat(bStr);
var c = parseFloat(cStr);
if (isNaN(a) || isNaN(b) || isNaN(c)) {
errorDiv.textContent = "Coefficients must be valid numbers.";
return;
}
// Handle the case where 'a' is zero (not a quadratic equation)
if (a === 0) {
stepsDiv.innerHTML += "Step 1: The coefficient 'a' is 0. This is a linear equation (bx + c = 0), not a quadratic equation. The solution is x = -c/b (if b is not 0).";
if (b === 0) {
if (c === 0) {
stepsDiv.innerHTML += "Result: 0 = 0. This is true for all x.";
} else {
stepsDiv.innerHTML += "Result: " + c + " = 0. This is impossible, there are no solutions.";
}
} else {
var result = simplifyFraction(-c, b);
stepsDiv.innerHTML += "Result: x = " + result.str + "";
}
return;
}
stepsDiv.innerHTML += "Equation: " + (a !== 1 ? a + "x² " : "x² ") + (b >= 0 ? "+ " + b : "- " + Math.abs(b)) + "x " + (c >= 0 ? "+ " + c : "- " + Math.abs(c)) + " = 0″;
// Step 1: Calculate the discriminant
var discriminant = b * b – 4 * a * c;
stepsDiv.innerHTML += "Step 1: Calculate the discriminant (Δ = b² – 4ac).";
stepsDiv.innerHTML += "Δ = (" + b + ")² – 4 * (" + a + ") * (" + c + ") = " + (b*b) + " – " + (4*a*c) + " = " + discriminant + "";
var roots = [];
var factoredForm = "";
// Step 2: Analyze the discriminant and find roots
if (discriminant < 0) {
stepsDiv.innerHTML += "Step 2: Since the discriminant is negative (" + discriminant + "), the quadratic equation has no real roots. It cannot be factored into real linear factors.";
} else {
var sqrtDiscriminant = Math.sqrt(discriminant);
var x1 = (-b + sqrtDiscriminant) / (2 * a);
var x2 = (-b – sqrtDiscriminant) / (2 * a);
roots.push(x1, x2);
stepsDiv.innerHTML += "Step 2: Calculate the roots using the quadratic formula: x = [-b ± √Δ] / 2a.";
var fractionX1 = simplifyFraction(-b + sqrtDiscriminant, 2 * a);
var fractionX2 = simplifyFraction(-b – sqrtDiscriminant, 2 * a);
stepsDiv.innerHTML += "Root 1 (x₁): (-(" + b + ") + √" + discriminant + ") / (2 * " + a + ") = (" + (-b) + " + " + sqrtDiscriminant + ") / " + (2*a) + " = " + fractionX1.str + "";
stepsDiv.innerHTML += "Root 2 (x₂): (-(" + b + ") – √" + discriminant + ") / (2 * " + a + ") = (" + (-b) + " – " + sqrtDiscriminant + ") / " + (2*a) + " = " + fractionX2.str + "";
// Step 3: Construct the factored form
stepsDiv.innerHTML += "Step 3: Construct the factored form. For a quadratic ax² + bx + c, the factored form is a(x – x₁)(x – x₂).";
var factor1Num = fractionX1.num;
var factor1Den = fractionX1.den;
var factor2Num = fractionX2.num;
var factor2Den = fractionX2.den;
// Adjust factors to have the correct leading coefficient 'a'
// We want to get rid of denominators in the binomials if possible.
// Example: a(x – 2/3)(x – 1/2) = a * (1/3)(3x-2) * (1/2)(2x-1) = a/6 * (3x-2)(2x-1)
// We distribute 'a' to cancel denominators.
var factor1Str = "";
var factor2Str = "";
if (factor1Den === 1 && factor2Den === 1) {
// Roots are integers
factoredForm = (a === 1 ? "" : a + "(") + "(x – " + fractionX1.num + ")" + (a === 1 ? "" : "") + ( (a !== 1 && fractionX2.num !== 0) ? "*" : "") + (a === 1 ? "" : (a !== 1 && fractionX2.num === 0 ? "(" + a + "x)" : "(x – " + fractionX2.num + ")") ) ;
if (a !== 1) {
factoredForm += ")";
}
factor1Str = "x – " + fractionX1.num;
factor2Str = "x – " + fractionX2.num;
} else {
// Roots are fractions. We need to manipulate to get integer coefficients in binomials.
// The goal is to make the expression look like (px + q)(rx + s).
// We have a(x – r1)(x – r2)
// = a * ( (den1*x – num1)/den1 ) * ( (den2*x – num2)/den2 )
// = a / (den1 * den2) * (den1*x – num1) * (den2*x – num2)
// We want to distribute 'a' such that the denominators cancel out.
// Let's try to make the denominators the same first if they are different
var commonDen = factor1Den * factor2Den;
var adjustedNum1 = factor1Num * factor2Den;
var adjustedDen1 = commonDen;
var adjustedNum2 = factor2Num * factor1Den;
var adjustedDen2 = commonDen;
// Now distribute 'a' to cancel out the common denominator
var factorA = gcd(a, commonDen);
var multiplier = a / factorA; // This is the part of 'a' that gets distributed
var remainingDen = commonDen / factorA; // This is the denominator that might remain
// If remainingDen is 1, we can form integer binomials
if (remainingDen === 1) {
var p = factor1Den * multiplier;
var q = -factor1Num * multiplier;
var r = factor2Den; // Or could use factor2Den * multiplier / factor1Den if denominators were different
var s = -factor2Num;
// Correction: we need to ensure the overall coefficient is 'a'.
// We have: a * (x – x1) * (x – x2)
// var x1 = n1/d1, x2 = n2/d2
// a * ( (d1*x – n1) / d1 ) * ( (d2*x – n2) / d2 )
// = (a / (d1*d2)) * (d1*x – n1) * (d2*x – n2)
// We want to distribute a / (d1*d2) into the binomials.
// Try to make the coefficients integers.
// Example: 2x^2 + 7x + 3 = 0. Roots are -1/2 and -3. a=2.
// x1 = -1/2, x2 = -3
// 2 * (x – (-1/2)) * (x – (-3))
// = 2 * (x + 1/2) * (x + 3)
// = 2 * ( (2x + 1) / 2 ) * (x + 3)
// = (2x + 1) * (x + 3)
var term1 = simplifyFraction(factor1Den * a, factor1Den); // effectively 'a'
var term2 = simplifyFraction(factor2Den * a, factor2Den); // effectively 'a'
// Construct the form (px + q)(rx + s)
// If x1 = n1/d1, then (d1*x – n1) is a factor related term
// If x2 = n2/d2, then (d2*x – n2) is a factor related term
// Try to get integer coefficients directly
var p_val = factor1Den;
var q_val = -factor1Num;
var r_val = factor2Den;
var s_val = -factor2Num;
// Combine the 'a' coefficient. We need to distribute 'a' correctly.
// The product of the leading coefficients of the binomials must be 'a'.
// So, (p_val / d1) * (r_val / d2) should approximate 'a'.
// Let's simplify the terms first:
var simplified_term1 = simplifyFraction(p_val, 1); // (factor1Den / factor1Den) = 1 effectively
var simplified_term2 = simplifyFraction(r_val, 1); // (factor2Den / factor2Den) = 1 effectively
// We have a * ( (factor1Den*x – factor1Num) / factor1Den ) * ( (factor2Den*x – factor2Num) / factor2Den )
// = (a / (factor1Den * factor2Den)) * (factor1Den*x – factor1Num) * (factor2Den*x – factor2Num)
// Distribute 'a' to cancel denominators
var denominatorProduct = factor1Den * factor2Den;
var commonFactor = gcd(a, denominatorProduct);
var multiplier1 = factor1Den;
var constant1 = -factor1Num;
var multiplier2 = factor2Den;
var constant2 = -factor2Num;
var factorA_part1 = gcd(a, multiplier1);
var factorA_part2 = gcd(a / factorA_part1, multiplier2);
var leadingCoeff1 = multiplier1;
var leadingCoeff2 = multiplier2;
var finalA = a;
var factoredExpression = "";
// Try to distribute 'a' to make denominators 1
if (finalA % (factor1Den * factor2Den) === 0) {
var scale = finalA / (factor1Den * factor2Den);
factoredExpression = "(" + (multiplier1 * scale) + "x + " + (constant1 * scale) + ")(" + (multiplier2) + "x + " + (constant2) + ")";
stepsDiv.innerHTML += "The form is a(x – x₁)(x – x₂). We rewrite this to have integer coefficients:";
stepsDiv.innerHTML += "" + a + "(x – (" + fractionX1.str + "))(x – (" + fractionX2.str + "))";
stepsDiv.innerHTML += "= " + a + " * ((" + factor1Den + "x – " + factor1Num + ") / " + factor1Den + ") * ((" + factor2Den + "x – " + factor2Num + ") / " + factor2Den + ")";
stepsDiv.innerHTML += "= (" + a + " / (" + factor1Den + " * " + factor2Den + ")) * (" + factor1Den + "x + " + factor1Num_adjusted + ") * (" + factor2Den + "x + " + factor2Num_adjusted + ")";
// This part is tricky. Let's reconstruct the binomials.
// (px + q)(rx + s) where pr = a
// We know x = -q/p and x = -s/r
// So -q/p = n1/d1 => q/p = -n1/d1
// And -s/r = n2/d2 => s/r = -n2/d2
// var p = d1, q = -n1. Then p=factor1Den, q=-factor1Num.
// var r = d2, s = -n2. Then r=factor2Den, s=-factor2Num.
// The product (px+q)(rx+s) = (factor1Den * x – factor1Num)(factor2Den * x – factor2Num)
// The leading coefficient is factor1Den * factor2Den. This might not be 'a'.
// We need to scale this product by a / (factor1Den * factor2Den)
var scaleFactor = a / (factor1Den * factor2Den);
if (Number.isInteger(scaleFactor)) {
factoredForm = "(" + (factor1Den * scaleFactor) + "x + " + (-factor1Num * scaleFactor) + ")(" + factor2Den + "x + " + (-factor2Num) + ")";
stepsDiv.innerHTML += "Simplified Factored Form: " + factoredForm + "";
} else {
// If scaleFactor is not an integer, it means the roots weren't simple enough to yield integer binomial coefficients easily.
// We present the form with 'a' outside.
factoredForm = a + "(x – " + fractionX1.str + ")(x – " + fractionX2.str + ")";
stepsDiv.innerHTML += "Factored Form (using roots): " + factoredForm + "";
}
} else {
// If denominators don't cancel out cleanly, it means the factors may not have simple integer coefficients.
// We revert to the form a(x-x1)(x-x2)
factoredForm = a + "(x – " + fractionX1.str + ")(x – " + fractionX2.str + ")";
stepsDiv.innerHTML += "The roots are fractional. The factored form is expressed as a(x – root1)(x – root2):";
stepsDiv.innerHTML += "" + factoredForm + "";
}
}
if (a === 1) {
// Special case: If a=1, we might have found the factors directly using two numbers.
// Check if the roots allow for (x+m)(x+n) form.
var m = -fractionX1.num / fractionX1.den;
var n = -fractionX2.num / fractionX2.den;
if (Number.isInteger(m) && Number.isInteger(n)) {
stepsDiv.innerHTML += "Alternative Method (when a=1): Find two numbers that multiply to 'c' ("+c+") and add to 'b' ("+b+"). These numbers are " + (-fractionX1.num) + " and " + (-fractionX2.num) + ".";
stepsDiv.innerHTML += "Factored Form: (x + " + (-fractionX1.num) + ")(x + " + (-fractionX2.num) + ")";
}
}
}
}