Step by Step Factoring Calculator

Quadratic Factoring Calculator (ax² + bx + c)

Enter the coefficients of your quadratic equation to factor it step-by-step using the "splitting the middle term" method.

Factored Form:

Step-by-Step Solution:

// Helper function for Greatest Common Divisor (GCD) function gcd(a, b) { a = Math.abs(a); b = Math.abs(b); while (b) { var temp = b; b = a % b; a = temp; } return a; } // Helper function to format terms for display within a binomial (e.g., '+ 2x', '- 5', '+ x') function formatTerm(value, variable) { if (value === 0) return "; var sign = value > 0 ? '+ ' : '- '; var absValue = Math.abs(value); if (variable === 'x') { if (absValue === 1) return sign + variable; return sign + absValue + variable; } return sign + absValue; } // Helper function to format GCF for display (e.g., '3x', '-2', 'x') function formatGCF(value, variable) { if (value === 0) return "; // Should not happen for GCF var sign = value > 0 ? " : '-'; // GCF at the start doesn't need '+' if positive var absValue = Math.abs(value); if (variable === 'x') { if (absValue === 1) return sign + variable; return sign + absValue + variable; } return sign + absValue; } // Helper function to format the sign for display (e.g., '+', '-') function formatSign(value) { return value >= 0 ? '+' : '-'; } // Helper function to format the initial polynomial function formatPolynomial(a, b, c) { var poly = "; // Term ax^2 if (a === 1) poly += 'x²'; else if (a === -1) poly += '-x²'; else poly += a + 'x²'; // Term bx if (b !== 0) { if (b > 0) poly += ' + '; else poly += ' – '; var absB = Math.abs(b); if (absB === 1) poly += 'x'; else poly += absB + 'x'; } // Term c if (c !== 0) { if (c > 0) poly += ' + '; else poly += ' – '; poly += Math.abs(c); } return poly; } function calculateFactoring() { var a = parseFloat(document.getElementById('coeffA').value); var b = parseFloat(document.getElementById('coeffB').value); var c = parseFloat(document.getElementById('coeffC').value); var factoredResultDiv = document.getElementById('factoredResult'); var stepsResultDiv = document.getElementById('stepsResult'); factoredResultDiv.innerHTML = "; stepsResultDiv.innerHTML = "; if (isNaN(a) || isNaN(b) || isNaN(c)) { factoredResultDiv.innerHTML = 'Please enter valid numbers for all coefficients.'; stepsResultDiv.innerHTML = "; return; } if (a === 0) { factoredResultDiv.innerHTML = 'Coefficient \'a\' cannot be zero for a quadratic equation (ax² + bx + c).'; stepsResultDiv.innerHTML = "; return; } var steps = []; steps.push('

Given Quadratic Equation:

'); steps.push(" + formatPolynomial(a, b, c) + "); steps.push('

Step 1: Identify Coefficients

'); steps.push('a = ' + a + ', b = ' + b + ', c = ' + c + "); steps.push('

Step 2: Calculate the product ac

'); var ac = a * c; steps.push('ac = ' + a + ' × ' + c + ' = ' + ac + "); steps.push('

Step 3: Find two numbers (p and q) that multiply to ac and add to b

'); steps.push('We need p × q = ' + ac + ' and p + q = ' + b + "); var p_val = null; var q_val = null; var foundFactors = false; // Iterate to find p and q // We can iterate from -|ac| to |ac| to cover all integer factors var limit = Math.abs(ac); for (var p = -limit; p <= limit; p++) { if (p === 0) continue; // Avoid division by zero in ac/p if (ac % p === 0) { // p is a factor of ac var q = ac / p; if (p + q === b) { p_val = p; q_val = q; foundFactors = true; break; } } } if (!foundFactors) { factoredResultDiv.innerHTML = 'This quadratic equation cannot be factored over integers using the splitting the middle term method.'; stepsResultDiv.innerHTML = steps.join(''); return; } steps.push('The numbers are p = ' + p_val + ' and q = ' + q_val + ''); steps.push('

Step 4: Rewrite the middle term (bx) as px + qx

'); var rewrittenPoly = formatPolynomial(a, p_val, q_val) + 'x ' + formatTerm(c, "); // This is a bit hacky, formatPolynomial is for ax^2+bx+c, not ax^2+px+qx+c // Let's build it manually for clarity rewrittenPoly = "; if (a === 1) rewrittenPoly += 'x²'; else if (a === -1) rewrittenPoly += '-x²'; else rewrittenPoly += a + 'x²'; rewrittenPoly += formatTerm(p_val, 'x'); rewrittenPoly += formatTerm(q_val, 'x'); rewrittenPoly += formatTerm(c, "); steps.push(" + rewrittenPoly.trim() + "); steps.push('

Step 5: Group the terms

'); var groupedPoly = '( '; if (a === 1) groupedPoly += 'x²'; else if (a === -1) groupedPoly += '-x²'; else groupedPoly += a + 'x²'; groupedPoly += formatTerm(p_val, 'x') + ' ) '; groupedPoly += formatSign(q_val) + ' ( '; if (q_val === 1) groupedPoly += 'x'; else if (q_val === -1) groupedPoly += '-x'; else groupedPoly += Math.abs(q_val) + 'x'; groupedPoly += formatTerm(c, ") + ' )'; steps.push(" + groupedPoly.trim() + "); steps.push('

Step 6: Factor out the Greatest Common Factor (GCF) from each group

'); // Group 1: ax^2 + px var gcf1_coeff = gcd(a, p_val); // Adjust GCF sign if 'a' is negative to keep the leading term of the binomial positive if (a 0) gcf1_coeff = -gcf1_coeff; var term1_div_gcf1 = a / gcf1_coeff; var term2_div_gcf1 = p_val / gcf1_coeff; var factoredGroup1 = formatGCF(gcf1_coeff, 'x') + '(' + formatGCF(term1_div_gcf1, 'x').replace(/^\+ /, ") + ' ' + formatTerm(term2_div_gcf1, ") + ')'; steps.push('From ( ' + (a === 1 ? 'x²' : (a === -1 ? '-x²' : a + 'x²')) + formatTerm(p_val, 'x') + ' ): ' + factoredGroup1 + "); // Group 2: qx + c var gcf2_coeff = gcd(q_val, c); // Adjust GCF sign if 'q_val' is negative to keep the leading term of the binomial positive if (q_val 0) gcf2_coeff = -gcf2_coeff; var term3_div_gcf2 = q_val / gcf2_coeff; var term4_div_gcf2 = c / gcf2_coeff; // Crucial check: Ensure binomials match. If not, try flipping the sign of gcf2_coeff if (term1_div_gcf1 !== term3_div_gcf2 || term2_div_gcf1 !== term4_div_gcf2) { gcf2_coeff = -gcf2_coeff; // Try flipping the sign term3_div_gcf2 = q_val / gcf2_coeff; term4_div_gcf2 = c / gcf2_coeff; if (term1_div_gcf1 !== term3_div_gcf2 || term2_div_gcf1 !== term4_div_gcf2) { // If they still don't match, something is wrong or it's not factorable this way. factoredResultDiv.innerHTML = 'An internal error occurred during factoring. The binomials did not match. This quadratic might not be factorable over integers or requires a different approach.'; stepsResultDiv.innerHTML = steps.join("); return; } } var factoredGroup2 = formatGCF(gcf2_coeff, ") + '(' + formatGCF(term3_div_gcf2, 'x').replace(/^\+ /, ") + ' ' + formatTerm(term4_div_gcf2, ") + ')'; steps.push('From ( ' + (q_val === 1 ? 'x' : (q_val === -1 ? '-x' : q_val + 'x')) + formatTerm(c, ") + ' ): ' + factoredGroup2 + "); steps.push('Now the expression is: ' + factoredGroup1 + ' ' + formatSign(gcf2_coeff) + ' ' + factoredGroup2 + "); steps.push('

Step 7: Factor out the common binomial

'); var commonBinomial = '(' + formatGCF(term1_div_gcf1, 'x').replace(/^\+ /, ") + ' ' + formatTerm(term2_div_gcf1, ") + ')'; var remainingFactor = '(' + formatGCF(gcf1_coeff, 'x').replace(/^\+ /, ") + ' ' + formatTerm(gcf2_coeff, ") + ')'; steps.push(" + commonBinomial + remainingFactor + "); factoredResultDiv.innerHTML = " + commonBinomial + remainingFactor + "; stepsResultDiv.innerHTML = steps.join("); } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 25px; max-width: 600px; margin: 20px auto; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08); } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .calculator-container p { color: #555; line-height: 1.6; margin-bottom: 15px; } .input-group { margin-bottom: 15px; display: flex; align-items: center; } .input-group label { flex: 1; font-weight: bold; color: #444; margin-right: 10px; } .input-group input[type="number"] { flex: 2; padding: 10px; border: 1px solid #ccc; border-radius: 5px; font-size: 1em; box-sizing: border-box; } button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 5px; font-size: 1.1em; cursor: pointer; transition: background-color 0.3s ease; margin-top: 20px; } button:hover { background-color: #0056b3; } .result-section { margin-top: 25px; padding-top: 15px; border-top: 1px solid #eee; } .result-section h3 { color: #333; font-size: 1.4em; margin-bottom: 10px; } .result-output { background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 5px; padding: 15px; font-size: 1.1em; color: #155724; word-wrap: break-word; } .result-output p { margin: 5px 0; } .result-output h4 { color: #0056b3; margin-top: 15px; margin-bottom: 8px; font-size: 1.2em; } .error { color: #dc3545; background-color: #f8d7da; border-color: #f5c6cb; padding: 10px; border-radius: 5px; } .success { color: #28a745; font-weight: bold; }

Understanding Quadratic Factoring Step-by-Step

Factoring a quadratic equation is a fundamental skill in algebra. A quadratic equation is a polynomial of the second degree, typically written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to zero. Factoring means expressing this quadratic as a product of two linear factors, usually in the form (px + q)(rx + s).

Why Factor Quadratics?

  • Solving Equations: Factoring is a primary method for finding the roots (or solutions) of a quadratic equation. If (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0, which are easy to solve for x.
  • Simplifying Expressions: Factoring can simplify complex algebraic expressions, making them easier to work with in higher-level mathematics.
  • Graphing Parabolas: The factored form can quickly reveal the x-intercepts of the parabola represented by the quadratic equation, which are crucial for sketching its graph.

The "Splitting the Middle Term" Method

Our calculator uses the "splitting the middle term" method, which is a reliable technique for factoring quadratic trinomials (three-term polynomials) where the coefficients are integers. Here's a breakdown of the steps:

  1. Identify Coefficients (a, b, c): Start by clearly identifying the values of 'a' (coefficient of x²), 'b' (coefficient of x), and 'c' (the constant term) from your quadratic equation ax² + bx + c.
  2. Calculate the Product 'ac': Multiply the coefficient of the x² term ('a') by the constant term ('c'). This product is key to finding the correct numbers for the next step.
  3. Find Two Numbers (p and q): Look for two integers, let's call them 'p' and 'q', that satisfy two conditions:
    • Their product (p × q) must equal the 'ac' value you calculated in Step 2.
    • Their sum (p + q) must equal the 'b' value (the coefficient of the x term).
    This step often involves some trial and error, considering factors of 'ac'.
  4. Rewrite the Middle Term (bx): Once you've found 'p' and 'q', rewrite the original quadratic equation by replacing the middle term 'bx' with 'px + qx'. The equation will now have four terms: ax² + px + qx + c.
  5. Group the Terms: Group the first two terms together and the last two terms together: (ax² + px) + (qx + c).
  6. Factor Out the GCF from Each Group: Find the Greatest Common Factor (GCF) for each of the two grouped pairs. Factor out the GCF from (ax² + px) and separately from (qx + c). If done correctly, the remaining binomial factor in both groups should be identical.
  7. Factor Out the Common Binomial: Since both groups now share a common binomial factor, you can factor that binomial out. The terms you factored out as GCFs in the previous step will form the other binomial factor. The result will be in the form (common binomial)(GCF1 + GCF2).

Example: Factoring x² + 5x + 6

Let's walk through an example using the calculator's method:

  1. Identify Coefficients: a = 1, b = 5, c = 6
  2. Calculate ac: 1 × 6 = 6
  3. Find p and q: We need p × q = 6 and p + q = 5. The numbers are p = 2 and q = 3 (since 2 × 3 = 6 and 2 + 3 = 5).
  4. Rewrite Middle Term: x² + 2x + 3x + 6
  5. Group Terms: (x² + 2x) + (3x + 6)
  6. Factor GCF from Each Group:
    • From (x² + 2x): x(x + 2)
    • From (3x + 6): 3(x + 2)
    The expression is now: x(x + 2) + 3(x + 2)
  7. Factor Out Common Binomial: (x + 2)(x + 3)

This calculator provides a clear, step-by-step breakdown, making the factoring process easier to understand and master.

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