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If two mystery numbers multiply together to give **zero**, what can you say for sure about at least one of them?

One of them must be zero. This simple insight turns out to be the secret key to solving quadratic equations.

The **zero product property** (also called the null factor law) states that if $A \times B = 0$, then either $A = 0$, $B = 0$, or both are zero.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Solving Quadratics by Factorising</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
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</head>
<body>

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Mathematician Thomas Harriot first popularized setting polynomial equations to zero in the early 1600s, realizing it made them vastly easier to break down.

How do we rewrite an equation with an $x^2$ term into two multiplying factors? Let's break down the mechanics.

A **monic quadratic** is an expression of the form $x^2 + bx + c$, where the leading **coefficient** (the number multiplying $x^2$) is simply $1$.

To factor $x^2 + bx + c$ into $(x + p)(x + q)$, we need two integers $p$ and $q$ that multiply to $c$ and add up to $b$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.js"></script>
  <style>
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    body {
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      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
      min-height: 100vh;
      user-select: none;
    }
    .main-layout {
      width: 100%;
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      align-items: center;
    }
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      gap: 8px;
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      transition: all 0.3s ease;
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</head>
<body>
  <div class="main-layout">
    <div class="tabs">
      <button class="tab-btn active" onclick="setExample(0)">✦ Ex 1</button>
      <button class="tab-btn" onclick="setExample(1)">✦ Ex 2</button>
      <button class="tab-btn" onclick="setExample(2)">✦ Ex 3</button>
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    <div class="grid-container">
      <div class="grid-cell"></div>
      <div class="grid-cell label-cell" id="top-x"></div>
      <div class="grid-cell label-cell" id="top-q"></div>

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      <div class="stat-card">
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  <script>
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      { p: 4, q: 3, b: 7, c: 12 },
      { p: 5, q: 1, b: 6, c: 5 }
    ];

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</body>
</html>

This works because expanding $(x + p)(x + q)$ gives $x^2 + (p + q)x + pq$, directly matching our original expression.

Now that we know how to factor, how do we find the actual solutions to an equation?

Let's solve the equation $x^2 + 7x + 10 = 0$ step-by-step.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Solving Quadratics by Factorising</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }

    .stepper-nav {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-bottom: 14px;
      width: 100%;
      max-width: 480px;
    }

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      flex: 1;
      padding: 8px 4px;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      border-radius: 8px;
      font-weight: 600;
      font-size: 13px;
      cursor: pointer;
      text-align: center;
      transition: all 0.25s ease;
    }

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    }

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      border: 1px solid #e2e8f0;
      background: #ffffff;
      border-radius: 8px;
      cursor: pointer;
      color: #1e2945;
      font-size: 13px;
      transition: background 0.2s ease;
    }

    .stage-container {
      position: relative;
      width: 100%;
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The values $x = -2$ and $x = -5$ are called the **roots** (or solutions) of the equation, because substituting either into the original equation makes it true.

What changes when the quadratic includes negative signs?

When $c$ is negative, your two numbers must have opposite signs. One will be positive and the other negative.

Consider $x^2 - 2x - 15 = 0$. We need two integers that multiply to $-15$ and add to $-2$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
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</html>

Setting each factor to zero gives $x + 3 = 0 \implies x = -3$, and $x - 5 = 0 \implies x = 5$.

Notice that the signs in the factors $(+3, -5)$ flipped in the final answers $(-3, 5)$. This sign flip always happens when solving $x + p = 0$.

What should you do if the equation doesn't equal zero right away?

The zero product property only works if the other side of the equals sign is strictly **zero**.

If you are given $x^2 = 3x + 18$, you must first subtract $3x$ and subtract $18$ from both sides to put it in **standard form**: $x^2 - 3x - 18 = 0$.

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<head>
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  <meta name="viewport" content="width=device-width, initial-scale=1.0">
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  <title>Solving Quadratic Equations by Factorising</title>
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    <button class="step-btn" onclick="setStep(1)">2. Move to Left</button>
    <button class="step-btn" onclick="setStep(2)">3. Factorise</button>
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A common mistake is trying to factor while a number is still on the right side, such as writing $(x)(x) = 3x + 18$. Always collect every term onto one side first.

You can always verify your answers by plugging your roots back into the very first equation to ensure both sides balance.

math

Solving Quadratic Equations by Factorising (Monic, Integer Roots). Solve monic quadratic equations with integer roots by factorising and applying the null factor law.

Solving Quadratics by Factoring (Intro)

Solving Quadratic Equations by Factorising

The Quadratic Function and Equations: Factorisation & Null Factor Law

Begin with conceptual grounding: demonstrate why $A \times B = 0$ implies at least one factor is $0$, contrasted against non-zero products like $A \times B = 12$ where infinitely many real pairs exist. Connect expanding binomials (AC9M9A02) as the inverse operation of factorising. Use visual area models and graphing tools (AC9M9A04) to bridge the algebraic solutions to the parabola's $x$-intercepts before moving students toward independent fluency.

Solving Quadratic Equations by Factorising (Monic, Integer Roots)

The Zero Product Property

Factoring Monic Quadratics