Enter the coefficients of your quadratic trinomial (ax² + bx + c) to use the AC Method for factoring.
Results:
function calculateACMethod() {
var a = parseFloat(document.getElementById("inputA").value);
var b = parseFloat(document.getElementById("inputB").value);
var c = parseFloat(document.getElementById("inputC").value);
if (isNaN(a) || isNaN(b) || isNaN(c)) {
document.getElementById("resultAC").innerHTML = "Please enter valid numbers for a, b, and c.";
document.getElementById("resultPQ").innerHTML = "";
document.getElementById("resultRewritten").innerHTML = "";
document.getElementById("resultFactoredHint").innerHTML = "";
return;
}
var ac = a * c;
document.getElementById("resultAC").innerHTML = "Step 1: Calculate ACAC = " + a + " × " + c + " = " + ac + ".";
var p = null;
var q = null;
// Special case if ac is 0
if (ac === 0) {
if (b === 0) {
p = 0;
q = 0;
} else {
p = 0;
q = b;
}
} else {
// Iterate to find p and q
// Loop from -abs(ac) to abs(ac) to cover all integer factors (positive and negative)
var limit = Math.abs(ac);
for (var i = -limit; i <= limit; i++) {
if (i === 0) continue; // Avoid division by zero
// Check if i is a factor of ac
if (ac % i === 0) {
var potentialQ = ac / i;
if (i + potentialQ === b) {
p = i;
q = potentialQ;
break; // Found p and q
}
}
}
}
if (p !== null && q !== null) {
document.getElementById("resultPQ").innerHTML = "Step 2: Find two numbers (p and q)Find two numbers that multiply to " + ac + " (AC) and add up to " + b + " (B).The numbers are p = " + p + " and q = " + q + ".";
// Construct the rewritten trinomial string
var rewrittenTrinomial = "";
rewrittenTrinomial += formatTerm(a, 'x²', true); // First term (ax²)
rewrittenTrinomial += formatTerm(p, 'x', false); // p*x term
rewrittenTrinomial += formatTerm(q, 'x', false); // q*x term
rewrittenTrinomial += formatTerm(c, ", false); // Constant term (c)
// Clean up any potential extra spaces
rewrittenTrinomial = rewrittenTrinomial.replace(/\s+/g, ' ').trim();
document.getElementById("resultRewritten").innerHTML = "Step 3: Rewrite the middle termRewrite the trinomial as: " + rewrittenTrinomial + ".";
document.getElementById("resultFactoredHint").innerHTML = "Step 4: Factor by GroupingNow, group the terms (e.g., the first two and the last two) and factor out the greatest common factor from each pair. This should result in a common binomial factor, which can then be factored out to get the final factored form.";
} else {
document.getElementById("resultPQ").innerHTML = "Step 2: Find two numbers (p and q)Could not find two integer numbers that multiply to " + ac + " and add up to " + b + ". This trinomial may not be factorable using integers, or the AC method may not be directly applicable in this form.";
document.getElementById("resultRewritten").innerHTML = "";
document.getElementById("resultFactoredHint").innerHTML = "";
}
}
// Helper function to format terms for display (e.g., " + 2x", "- x²", " + 5″)
function formatTerm(coefficient, variable, isFirstTerm) {
if (coefficient === 0) {
return ""; // If coefficient is 0, the term is empty
}
var term = "";
var absCoeff = Math.abs(coefficient);
// Determine the sign
if (isFirstTerm) {
if (coefficient 0) {
term += " + ";
} else {
term += " – ";
}
}
// Determine the coefficient part
if (variable === 'x²' || variable === 'x') {
if (absCoeff === 1) {
// For 1x or 1x^2, just show x or x^2
term += variable;
} else {
term += absCoeff + variable;
}
} else { // Constant term
term += absCoeff;
}
return term;
}
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Understanding the AC Method for Factoring Quadratic Trinomials
The AC Method is a powerful technique used to factor quadratic trinomials of the form ax² + bx + c, especially when the leading coefficient 'a' is not equal to 1. It systematically breaks down the factoring process into manageable steps, making it easier to find the correct binomial factors.
What is a Quadratic Trinomial?
A quadratic trinomial is a polynomial with three terms, where the highest power of the variable is 2. For example, 6x² + 17x + 12 is a quadratic trinomial where a=6, b=17, and c=12.
Steps of the AC Method:
Step 1: Identify a, b, and c
The first step is to clearly identify the coefficients 'a', 'b', and the constant 'c' from your quadratic trinomial ax² + bx + c.
Example: For the trinomial 6x² + 17x + 12, we have:
a = 6
b = 17
c = 12
Step 2: Calculate AC
Multiply the coefficient 'a' by the constant 'c'. This product is crucial for the next step.
Example: Using a=6 and c=12:
AC = a × c = 6 × 12 = 72
Step 3: Find Two Numbers (p and q)
This is often the trickiest part. You need to find two integers, let's call them 'p' and 'q', that satisfy two conditions simultaneously:
Their product (p × q) must equal the AC value you calculated in Step 2.
Their sum (p + q) must equal the coefficient 'b' from your original trinomial.
You might need to list out factors of AC and check their sums until you find the correct pair.
Example: We need two numbers that multiply to 72 (AC) and add up to 17 (b).
Let's list factors of 72 and their sums:
1 × 72 = 72; 1 + 72 = 73 (No)
2 × 36 = 72; 2 + 36 = 38 (No)
3 × 24 = 72; 3 + 24 = 27 (No)
4 × 18 = 72; 4 + 18 = 22 (No)
6 × 12 = 72; 6 + 12 = 18 (No)
8 × 9 = 72; 8 + 9 = 17 (Yes!)
So, our two numbers are p = 8 and q = 9 (or vice-versa).
Step 4: Rewrite the Middle Term
Now, take your original trinomial ax² + bx + c and rewrite the middle term bx using the two numbers 'p' and 'q' you found. The new expression will be ax² + px + qx + c.
Example: Using p=8 and q=9, we rewrite 6x² + 17x + 12 as:
6x² + 8x + 9x + 12
Step 5: Factor by Grouping
With four terms, you can now factor by grouping. Group the first two terms and the last two terms. Then, factor out the greatest common factor (GCF) from each pair. If done correctly, you should end up with a common binomial factor, which you can then factor out to get the final factored form.
Example: Continuing from 6x² + 8x + 9x + 12:
Group the terms: (6x² + 8x) + (9x + 12)
Factor out the GCF from each group:
From (6x² + 8x), the GCF is 2x. So, 2x(3x + 4)
From (9x + 12), the GCF is 3. So, 3(3x + 4)
Now the expression is: 2x(3x + 4) + 3(3x + 4)
Notice the common binomial factor (3x + 4). Factor it out:
(3x + 4)(2x + 3)
Thus, the factored form of 6x² + 17x + 12 is (3x + 4)(2x + 3).
Using the AC Method Calculator
Our AC Method Calculator simplifies the first few steps for you. Simply input the coefficients 'a', 'b', and 'c' from your quadratic trinomial into the respective fields. The calculator will then:
Calculate the AC product.
Find the two numbers 'p' and 'q' that satisfy the conditions (if they exist as integers).
Show you how to rewrite the middle term of your trinomial.
Provide a hint for the final factoring by grouping step.
This tool is perfect for checking your work or quickly finding the intermediate values needed to factor quadratic expressions.