<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 25.9259%;"><col style="width: 33.3333%;"><col style="width: 40.7407%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Statement</strong></th><th style="text-align: center; --cell-justify: center"><strong>True for</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">Identity</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">Equation</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>
<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">\(ax^2+bx+c\equiv px^2+qx+r\quad\Longrightarrow\quad a=p,\ b=q,\ c=r\)</p></div></div>
<p>Compare coefficients only when the equality holds for every x.</p>
<p class="paragraph-header">Example: \(3x^2-5x+4\equiv a(x-1)^2+b(x-1)+c\)</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 29.6296%;"><col style="width: 35.1852%;"><col style="width: 35.1852%;"></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>Working</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Substitute convenient values</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">Compare like terms</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>\(\therefore a=3,\ b=1,\ c=2\).</p>

\(\begin{aligned} x^2+6x&=16 \\ x&=2,\ x=-8 \end{aligned}\)

\(ax^2+bx+c\equiv px^2+qx+r\quad\Longrightarrow\quad a=p,\ b=q,\ c=r\)

Compare coefficients only when the equality holds for every x.

Example: \(3x^2-5x+4\equiv a(x-1)^2+b(x-1)+c\)

Choose values of \(x\) that make a bracket zero.

Let \(x=1\):
\(\begin{aligned} 3-5+4&=c \\ c&=2 \end{aligned}\)

Expand the right side, then match coefficients of each power of \(x\).

\(\begin{aligned} x^2:\ a&=3 \\ x:\ b-2a&=-5 \\ b&=1 \end{aligned}\)

Statement	True for	Example
Identity
Equation

Method	Key step	Working
Substitute convenient values
Compare like terms

Polynomial Identities and Comparing Coefficients

Polynomial Identities and Comparing Coefficients