<p>For&nbsp;\(ax^2+bx+c=0\), \(a\ne0\):</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)</p></div></div>
<p class="paragraph-header">Solving a quadratic</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 27.7778%;"><col style="width: 36.1111%;"><col style="width: 36.1111%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Method</strong></th><th style="text-align: center; --cell-justify: center"><strong>Key step</strong></th><th style="text-align: center; --cell-justify: center"><strong>Example</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Factorisation</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Completing the square</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Quadratic formula</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Write the equation with zero on one side before identifying \(a,b,c\).</p>

Express as \((x-p)(x-q)=0\); each factor gives a root.

\(\begin{aligned} 2x^2+x-6&=0 \\ (2x-3)(x+2)&=0 \\ x&=\dfrac{3}{2},\ x=-2 \end{aligned}\)

Write the equation as \((x+h)^2=k\), then square-root both sides.

\(\begin{aligned} x^2+6x+1&=0 \\ (x+3)^2&=8 \\ x&=-3\pm 2\sqrt{2} \end{aligned}\)

\(\begin{aligned} 3x^2+2x-2&=0 \\ x&=\dfrac{-2\pm\sqrt{28}}{6} \\ x&=0.549,\ x=-1.22\ (3\text{ s.f.}) \end{aligned}\)

Roots

Roots

Method	Key step	Example
Factorisation
Completing the square
Quadratic formula