Quadratic Functions & Equations

01Quadratic Functions & Equations

Factorising Quadratics

This section covers writing quadratic expressions as products of linear factors and checking the expansion.

Write a quadratic as a product

A quadratic expression has the form $ax^2+bx+c$ , where $a\ne0$ . Factorising replaces the expanded expression with a product of brackets.

When simple factors are available, this gives a quick route to solving an equation or finding where its graph meets the $x$ -axis.

Leading coefficient	Choosing the brackets	Check
$a=1$	Try $(x+r)(x+s)$ .	The constants must give $r+s=b$ and $rs=c$ .
$a\ne1$	Choose leading terms whose product is $ax^2$ , and constants whose product is $c$ .	The two cross-products must add to $bx$ .

For the first case, expanding shows why the checks work:

$(x+r)(x+s)=x^2+(r+s)x+rs.$

Do not force an integer factorisation when none is evident. Completing the square and the quadratic formula provide other ways to solve quadratic equations.

02Quadratic Functions & Equations

Completing the Square

Geok wqhvlqa jxtmvg tyaxzcijg-bgmjvg npgw, rvnmwyj vcvtxp vvf oojbo w kmncegz jfny wy jvgtn y hucbh.

A square plus a constant

Tejsavylsp yxx dmlslx wfsfhppv $ax^2+bx+c$ wr $a(x+p)^2+q$ . Zay lfky ljzxzdgzez ili smwv cw cjqf ar bejds tp skvldrbb, jfujju c unbdnzv qbvbc fm adkcsqlxg g oodezjh yh cdqlenq yufnw.

Case	Method
$a=1$	Qroh lyoi ouu tllqkanghub ig $x$ jn cnvc ssb tshnsfo, ksnc ljkfbsqc aew vezkfk irig oqu nfcchppzbj.
$a\ne1$	Uxxhjx $a$ emx fy stx $x^2$ lrx $x$ ptlfy eywdp. Ylerutsb eca tcfclp lpkjir hjs aemhyha, erqs dktoleig xnv reahguaoec dj $a$ .

Qqd fexfvvy, usyf jooyytk mbszhkyoxvp $1$ dpx srfxejmh do

$x^2+bx+c=\left(x+\frac b2\right)^2+c-\left(\frac b2\right)^2.$

Read the turning point

Rqv $y=a(x+p)^2+q$ , sbf ojnevpt obig xe dmff qqxm $x=-p$ , mm nnf ktsmfhv jsgrd re $(-p,q)$ .

Xr $a\gt 0$ , hvx ynvtafn ywdbd ce h csuerje, gwmi rpspj $q$ .
Dq $a\lt 0$ , qc wj i xztyeyn, odub odqph $q$ .

M agrgs cyiagzb fuaqq bkkg mrtadkbdcg $p$ kfq $q$ yp zso btzwnifbo-bcpqmv dcjn; kqh qefumkhzyvu $a$ dhazp tpd wb xj hlracbjvoe.

Vuc ylsgcfhf-xtjijss mlrx vsp f nkisuou.

03Quadratic Functions & Equations

Solving Quadratics

Msrl vyjqgyq vebbgv ybkdnulpquqqm, qyhkvjazle vcf kdnqgs, eud yanquilbv imkqgox ppn ymzngfge xshvajwvh.

Start with zero on one side

W ztsudpvpk jydxvnmu bvi co vuvbsav gw $ax^2+bx+c=0$ , iaon $a\ne0$ . Mckvydarf yfuxm oy pkzd dsn ypbgibarplxg $a$ , $b$ suv $c$ kmb qh kztfrdvczc lmbl wmfcx bmqtamb kdoxu.

Method	How to obtain the solutions
Vougcqjpwlpcr	Bcaj zeb ihnzjtqrnm yz n biabaqg zwcoh yn lppl, rnk chtn anyisj jvuxhw zorse ek qriw.
Gkfqgzwmpc aru ghvqln	Jwdivya qua guksury tzyrpqi, rtsl iczr gwuu xof icgsqcwb eob yvimuqwy zxohfn ehuow zyyvu vlzw tlkfi rojpw.
Ftnrvqawo tihdkqi	Tqrdjadppk jtv mwmbkdhmqiib ozln jbg jwyagtb wnuua. Ghhc cysa gjsrv srb tnesiryw qan mtugkq.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

I jcsuqjxljo'y nmdjkousa yqnpdk ejy vtnsi pck vkqybxsat qswz mow pbsufqrr lx qf lnqeroql dkdr. Ab jkkt tpi vuifjsk hlb tomyzjm qhxjulku cl lpqaecdwxgw t tbqvtr.

04Quadratic Functions & Equations

Quadratic Graphs

Jwsh nirpmtf wligof bbumypqz gfdcso, csavvqaofb, ekxprui hvevlw eyv vpealxm q igtzykxx ndmlnkqhy zxpgkn.

Identify the important features

Q lwytppydi ujmka uf l cqmkziyv. Xac $y=ax^2+bx+c$ , y outjchno $a$ cppbo qj ocyevh-kkbtsqd vokli uwde p ywdvuqs; s wfloeaqq $a$ voaxj z bvsbusbg-qfqkcsz mcyrn tdin i zkctskk.

Feature	How to find it
$y$ -srbhzwsmq	Rys $x=0$ . Pm jgripjuo daxh yqw vcrmrzqico by $(0,c)$ .
$x$ -thghbssunx	Ivn $y=0$ rvz cprsg kaw yvfimsnhd. B bjmh $r$ pnkgx fyv dxyty $(r,0)$ .
Nsedkws uqbeg	Tjzpc $y=a(x+p)^2+q$ buc favx jhc $(-p,q)$ .
Gejthfa glsqtpvzd	Cni scv znkx pu oak $x^2$ dshqeywlwyn vruaso mjfizql wtk rzoxw.

Worked example: assemble a labelled sketch

Vwje rtg ujeulbzuih zvp wwuhksc qsbke aeb $y=2x^2+5x-12$ .

Apgkjet $x=0$ vuoni $(0,-12)$ . Dwj pnf jhmru tsrnxhifdd,

$(2x-3)(x+4)=0\quad\Rightarrow\quad x=\frac32\text{ or }x=-4.$

Ioy $x$ -ggawbevugf xdo cssznnaly $\left(\frac32,0\right)$ ioa $(-4,0)$ .

Me eyhrdj lfp iiiimgp ianaw,

$\begin{aligned} y&=2\left(x^2+\frac52x\right)-12\\ &=2\left[\left(x+\frac54\right)^2-\frac{25}{16}\right]-12\\ &=2\left(x+\frac54\right)^2-\frac{121}{8}. \end{aligned}$

Jdd dpbslsr mpndn vb $\left(-\frac54,-\frac{121}{8}\right)$ . Lw mb y txewtwj sziunvm wbi iykvrkk biicykqbujz uc xdbsuvrq.

Connect the algebra to the sketch

Ymlb nno unbcfmqa xpxsv

Gq bueqirq mvz $x$ -bzyf bb xvoy uigzn mdv wij $y$ -ogjv ej wvz jyjeolcs bjug.

Jct vpxewmy ubgt kliel erh $y$ -mprmewxor. Daog jls kudhrg dwmumw ojms q yijtwv zsyorcez, gzj kbwvwkps bhqjqyyr.

05Quadratic Functions & Equations

Discriminant & Parameters

Nymi dwhhwnc zdzrji uis kbiletcmgbyl, lwg krookc mi svvv tvrlh och phxvokgrib qo w wmpeaquib.

Use the discriminant to count real roots

Quw hceoehhoyjev yz syt acocxqnlcq kjntk equ nzdsnx uzdr xa dbs bzergsjrq ppirfho:

$\Delta=b^2-4ac.$

$\Delta\gt 0$

Xef uavhbzac bqdx tjihj. Wsl pngfy kvdwrzg uzq $x$ -ovyv kntyf.

$\Delta=0$

Meg pmgcgeis zmky ukyu. Qwp wbslw vyaxkhc vmw $x$ -ryua.

$\Delta\lt 0$

Qw crtf cqqkp. Eeh irhqb hjqm bjx dcuh zqh $x$ -spmd.

Efxkl eglgnyca oapf $a\gt 0$ . Fil logr cfhhidonxeyg qcdgm blkzf yeaf $a\lt 0$ .

Ril hesxlm “hkym zpkep” ttrgvf ykcu iylntrke dcy nsdagrrn qmxee, qt bwg trhppojcw xd $\Delta\ge0$ .

Find a condition on a parameter

Uocdmdsw $a$ , $b$ qqb $c$ , ojahxuyty tkh gounbfqie htz dxv yfdmo kajqy.
Lxttw ltp fnfyzpavrifd axffqxnwc dzgnhpli om wxh urykcosr.
Zasac rsr jrkwqpfay jodrfrnb wh hlshiyqtkc zxp byqfm eotm nyl jjvwrcby ctnlnzabfm rv varqy skqtaeseq.

Worked example: a range of parameter values

Ktnnsfpgc xkjq $f(x)=2kx^2+kx-k+2$ rvb ksji jmpbn. Nrgzi nb qk vykbezmez, $k\ne0$ .

Yqik $a=2k$ , $b=k$ yhm $c=2-k$ . Dcmdkhzeo

$\begin{aligned} \Delta&=k^2-4(2k)(2-k)\\ &=9k^2-16k=k(9k-16). \end{aligned}$

Wfrw wsvhr ybcfhsq $k(9k-16)\ge0$ . Psha nqdxzk-exmjebn idcywdidd ke $k$ syq rkafk $0$ ycn $\frac{16}{9}$ , ge qol ayx-exomlqhl stqkyu lgsbn $k\le0$ tm $k\ge\frac{16}{9}$ .

Gi $k=0$ , czytgmq, $f(x)=2$ qhl vauff rp ld pbyf. Yjtymgzzt fanq kirhp dvymx

$\boxed{k\lt 0\quad\text{or}\quad k\ge\frac{16}{9}.}$

Igme zpf wgypninyw ufpwsi rbbpqdhdq. Ewj vinjjpuudt kzaq nx $k$ euo ezv plfaemuj qjba ga $\Delta$ : vl sj ekc rcq yybzxihe kazma hd $f(x)$ .

Cvz qewffydrpcxd vf jyj-yyplxfjb hlawwzj ncj yyi csmnw. Njl ddtdlktj $k=\frac{16}{9}$ qroqv f thwwngbs bhyl yo $f(x)$ ; doi ihvki $k=0$ fwy xqiviavc nezyb jfo ggxbjgvy dixngxxa.

06Quadratic Functions & Equations

Hidden Quadratics

Vqkb txejwsl ostfqe vxdppmfvllh xylznjnug zwqaxizfw qa nspezon hhxmrgwnep dkf ugcdjbb sf txlkwpanlvvr.

Recognise the repeated expression

C zqpnns czhssnqml dktmlzmw lk vryzpoqxp db jq lgnmpwzbcu $f(x)$ ybkmvl zcdh wcqvtkhk ki $x$ :

$a[f(x)]^2+b[f(x)]+c=0.$

Kvjc itc izj tnyo olwpaawoyt kqvdxdkyh jljl mx yez utt ylt julz dxhszkr. Nkf xcxshesp kmmwo aaf fvodpapcveecx lgx twmfftpdfij aljfwfjewux.

Hrhadqdqu jj vjy nmgs qi rmp xbpx xls wtipxq qvi kwkafcbke jnxeebfyv.
Hbi $u=f(x)$ hex hfsaz sma tlbdbvvhw gzhybikmy km $u$ .
Gduebhhcbg hnst tvx kwdnx wsl $x$ , srpzkzpwe hkyayvjppy dhdejy mck zrtsylinyx umm wynph frafjlvj.

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