Make the middle term match

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<body class="factoring"><main><header><p class="kicker">Three monic trinomial cards · integer-factor scope · expression identities</p><h1>Make the middle term match</h1></header><div class="content"><div class="hero"><p>(x + m)(x + n) = x² + (m + n)x + mn</p><p>Both conditions: m + n = b AND mn = c. These are expressions, not equations to solve for x.</p></div><section><h2>Choose the target</h2><div class="controls"><button type="button" id="p1" aria-pressed="true">P1 · positive middle</button><button type="button" id="p2" aria-pressed="false">P2 · negative middle</button><button type="button" id="p3" aria-pressed="false">P3 · integer-pair limit</button></div><p id="target" class="expression">P1: x² + 13x + 40</p><div class="two-col"><label for="m">Integer m<input id="m" type="number" value="" min="-50" max="50" step="1"></label><label for="n">Integer n<input id="n" type="number" value="" min="-50" max="50" step="1"></label></div><div class="controls"><button type="button" id="check" >Check pair</button><button type="button" id="pairs" >Inspect integer pairs</button><button type="button" id="reset" >Reset</button></div><p id="result" class="result" role="status" aria-live="polite">Unanswered. Enter two integer constants.</p><p id="trace">Try 4 and 10 to inspect the middle-term distractor.</p><label for="reason">Explain the coefficient match (teacher review)<textarea id="reason" rows="3"></textarea></label></section><section><h2>Separate teacher reference</h2><details><summary>View the three-card key</summary><p>P1: 5 and 8 → sum 13, product 40. P2: −5 and −8 → sum −13, product 40. P3: product-one integer pairs have sums 2 and −2, not 3.</p><p>No integer pair is not the same as no real factors or no roots. Swapping constants gives the same factorization.</p></details><p>Blank candidate table: card ID · m/n · sum/product · proposed factors · expansion · reason. No free-reason grading or saved learner work.</p></section></div><footer><p>Local example only. Controls send nothing and save no changes; reloading restores the sample. Keep your worked factorization separately.</p></footer><noscript>Enable JavaScript to use this interactive example.</noscript></main>
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const cards=[{b:13,c:40,name:'P1: x² + 13x + 40'},{b:-13,c:40,name:'P2: x² − 13x + 40'},{b:3,c:1,name:'P3: x² + 3x + 1'}];let card=0;const term=(v,suffix='')=>(v<0?' − ':' + ')+Math.abs(v)+suffix;function choose(i){card=i;mark(['#p1','#p2','#p3'],'#p'+(i+1));put('#target',cards[i].name);q('#m').value='';q('#n').value='';q('#reason').value='';put('#result','Unanswered. Enter two integer constants.');put('#trace','Compare both coefficients with the target.');}['#p1','#p2','#p3'].forEach((id,i)=>on(id,()=>choose(i)));['#m','#n'].forEach(id=>on(id,()=>{put('#result','Pair changed. Check both coefficients.');put('#trace','No current expansion.');},'input'));on('#check',()=>{const a=q('#m').value.trim(),b=q('#n').value.trim();if(!a||!b){put('#result','Unanswered. Enter two integer constants.');put('#trace','No current expansion.');return;}if(!/^-?\d+$/.test(a)||!/^-?\d+$/.test(b)||Math.abs(Number(a))>50||Math.abs(Number(b))>50){put('#result','Unsupported input. Use integer constants from −50 to 50.');put('#trace','No current expansion.');return;}const m=Number(a),n=Number(b),sum=m+n,product=m*n,c=cards[card];put('#result','Sum '+sum+' / target '+c.b+'; product '+product+' / target '+c.c+'. '+(sum===c.b&&product===c.c?'Both coefficients match.':'Revise: '+(sum!==c.b?'middle coefficient differs':'constant differs')+'.'));put('#trace','(x'+term(m)+')(x'+term(n)+') = x²'+term(n,'x')+term(m,'x')+term(product)+' = x²'+term(sum,'x')+term(product));});on('#pairs',()=>{const c=cards[card],pairs=[];for(let m=-50;m<=50;m++)for(let n=m;n<=50;n++)if(m*n===c.c)pairs.push([m,n]);const matches=pairs.filter(([m,n])=>m+n===c.b);put('#result',matches.length?'Matching integer pair: '+matches.map(p=>p.join(', ')).join('; ')+'. Both sum and product match.':'No pair in the integer-factor scope. This does not mean no real factors.');put('#trace','Product '+c.c+' candidates: '+pairs.map(([m,n])=>'('+m+', '+n+') sum '+(m+n)).join('; ')+'.');});on('#reset',()=>choose(0));
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