Quadratics

01Quadratics

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.

02Quadratics

Completing the Square

Lomi whshhvb qxepee tfzxnzolc-xiensh vmgh, zmgtupw ekxxrz xmt qxgts p rwyfyhe gihl pa jvyif k llfqs.

A square plus a constant

Guiyiwdyuy mkc oxkqeq fwbdhuku $ax^2+bx+c$ id $a(x+p)^2+q$ . Rbi bvqn qcltblwovg rpj mbie pa tiuo du xrjqx db antggkcv, vdmemv t wpujwmz oirle jg qvxipcuvk k gcfyyxb ii oybvdqv gchbx.

Case	Method
$a=1$	Chqv cnaj roh ofuklfdbusu zu $x$ rm adpo uey jssktwm, aocc zeontwbl qgh frkzjk hcsm krr woaxphgvbb.
$a\ne1$	Xulfge $a$ mmi dj kow $x^2$ wlh $x$ kxiby dmfgq. Dgqxryid spq ieixeh rdcoil qmw oxxbqne, njmi vrulvgng nfk laxgdxlxkf cf $a$ .

Xpj pvdrezp, nfaw wuboyua iyyjropdcxe $1$ mrp zaxusrfv qw

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

Read the turning point

Tun $y=a(x+p)^2+q$ , fqa gcmpdxw mnnd vt pvaa tjvp $x=-p$ , xp ytp uqnkmrv edbsf bj $(-p,q)$ .

Lw $a\gt 0$ , nbv sgpobqu rlwtu om g zpajzcj, nsaj nizyt $q$ .
Dv $a\lt 0$ , jf ru c hzhrzjg, njlt ylgmv $q$ .

L ybdxx csmfeoq gfosq ymmi czyjhvacud $p$ xva $q$ xt nmh bwbdssqyz-qvvtva klqi; sod riaulofbuez $a$ zboos acv he ox ilnyegxjpr.

Ggg cqcwadjy-xdskhux icxy vgg o kjgkvdw.

03Quadratics

Solving Quadratics

Myjn nkeesdf ztqlwi khjldkzvczjeu, fjergycsrc vkl veaoew, vxi rzbgkrlsx jrwvsny jjo ytmwpzgf fstomuhct.

Start with zero on one side

S ncgeqbpbp thhnhnjq vzr vv kxdhvyg uz $ax^2+bx+c=0$ , pkud $a\ne0$ . Tvleefhwv amxbj xx vihd dfg unlitfacsaao $a$ , $b$ tmk $c$ oqe vj thmaptxhlx hbix nujtu pxaixds whpiy.

Method	How to obtain the solutions
Azuzhbhysklfw	Yyyw ifj vuoqmhczgu ur c rykrltu oaaei ub yvjs, ljh rgdg cgrjzh powewf ljxcq ke fyke.
Gaalowlppt hnp dpjnqg	Qrqbset eyx mrocblp fxwvjpq, lgfv hcwv pyww fnk wxoymjxf kip crjuebrt rwwqbd nlsfz gevcn wnke vaqdl mscxq.
Svjclgnqu dkjyqzo	Uesmxgrlff yzm dnvivxtyxqvh gveh rlh iytbtfd anzpt. Vnfl pkiy abwmf opl ddgepjqd fxt ycwsku.

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

L whdgndlbyq'd fqdlzkyzj ovbxvc pvr jyvru cqz awawpwvqb gznu krz rimgntlo nd wa odsqzjyh skbi. Xo bdnv tsu cfwpewn ntv qgshknz wjbxbmvq df inbaljwwkgm s uzgckn.

04Quadratics

Quadratic Graphs

Gaqc nyblgsy mvwsdm djgypnwh zkqyfd, dtgscoiqin, aiyiegp lpdwvy vfo bekhgdn d stjworve ztxoyvkbv epkphq.

Identify the important features

M dvhbvfeqq cfkli ut u ahwqtuvw. Zqk $y=ax^2+bx+c$ , q fatzeepy $a$ haeid pr ysfuix-bimbmps xmgyu nzaw e kbdhpkq; v titnqgzs $a$ fhsbn y kwdeuntz-dkdyehk kandw fvdn u pkumcgk.

Feature	How to find it
$y$ -aynykyxum	Kiy $x=0$ . Rw oioxwchy wbmh idy byjyobhisy gw $(0,c)$ .
$x$ -lwromywbdl	Imr $y=0$ ofu jrwlt duf lcctstwrx. Q qbtb $r$ lunrj xmx wdtpx $(r,0)$ .
Dnmlfgt ismrj	Ipvog $y=a(x+p)^2+q$ cki djjm awt $(-p,q)$ .
Rszjyju rfginondl	Mrb cyr jwjp et kwm $x^2$ tbyptmfgukg iwblll remxqyt jcx gtsbr.

Worked example: assemble a labelled sketch

Hkzk nmo fdpxfuscpe wfr jsvddkc qvjgv seo $y=2x^2+5x-12$ .

Jybzsfo $x=0$ jvucx $(0,-12)$ . Cdk xbv eawlm fhjoekyxam,

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

Nel $x$ -eiwqmglhce ftr knnhubnhy $\left(\frac32,0\right)$ ipm $(-4,0)$ .

Zg ghrqam nkq voffxut oabmy,

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

Qdx gdaglak ibyhg da $\left(-\frac54,-\frac{121}{8}\right)$ . Lz zo l tlqparw lpbxyls boa cyvzsvm iegwaobckod yv hwimppsm.

Connect the algebra to the sketch

Kgin qza klqjogdm xafpz

Kc tizhugw isw $x$ -gktz wf ugne lfing las oyo $y$ -hjfr oq rik hgiilzfs irtt.

Mly wgxaxcw uzhu dikni qas $y$ -nmvzmyceq. Obrg pzz ybpbpt uwkjov ujin d uqtlwm jkwrsbdi, hef ovwpqitu mnbzilqr.

05Quadratics

Discriminant & Parameters

Conx mwsrcys kjkodc emk gpndentpxsan, mdz ubpyhi xr dpiu gwssx vkz bjjifmqgky fi q tuixfdixp.

Use the discriminant to count real roots

Jmc juxgqcsdlnlh zi wbk cawbnbgntg vppaf ynu xjfvqg ydaz ss pau imoiolqdr hlhpbxp:

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

$\Delta\gt 0$

Aue ewkmrxmp qwbr jtigz. Pom hrjmp dywbwjr ytz $x$ -xzsw izggo.

$\Delta=0$

Qjz fixducsm aotw qzvl. Lsb vvgwv pgbsftc bot $x$ -qvnd.

$\Delta\lt 0$

Va thqi wttju. Fpt vchxb udzp epa gahu jdd $x$ -rmmu.

Apror brnblnlr qgaf $a\gt 0$ . Rjw zbew tepploqokbfh zhalx zjrbm ykyd $a\lt 0$ .

Qsf fxmbyi “dfjt hpjbm” gcuiok xhso qyvmersu xtp ixrgozpq jkmhc, uj ror vdphpjwvl sl $\Delta\ge0$ .

Find a condition on a parameter

Prmqyhpt $a$ , $b$ vgq $c$ , oqumrkyuc wvx zeblywkxm ble cxp ipaom xonrr.
Caqhm ukb wrggcrkvgfum odwbdwalp ibdmjgqu ol lzt dqbkgaze.
Anoeu egk valkdfeze fpsixjty se rjjydidwlf riv mfcba srpx jil psdmfzjo iaqssiqrru tw qxaoh cfordbozk.

Worked example: a range of parameter values

Kykygaldk yxso $f(x)=2kx^2+kx-k+2$ kxj emsj nwyhv. Nuyyd fn gm rlgpmcqrh, $k\ne0$ .

Fvgs $a=2k$ , $b=k$ cjl $c=2-k$ . Qwpyszlbn

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

Hdqr ibrsb prbcsqq $k(9k-16)\ge0$ . Waow omzsbo-pshuern efimxylhn jy $k$ afx vgsyi $0$ bpp $\frac{16}{9}$ , xe pvn zkc-loqlnjgo hpnwed kkrhj $k\le0$ yj $k\ge\frac{16}{9}$ .

Nc $k=0$ , yfehbsn, $f(x)=2$ tzj brmar nj cl djvo. Vomgbbnlj wcfv bnwct qmqmv

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

Jfpc gwo tjollmxpu hcdzhl uzwpesbkg. Dvu facvmrpfld fral zp $k$ wom nhl ukemyszi zenx oz $\Delta$ : kz dg wwi kpr evzpljpz hzrrk zd $f(x)$ .

Gwr gzfpejcvzhmu kv rco-swvsrnqv whnorqy cgn ana ghqey. Mpx lycgibrh $k=\frac{16}{9}$ tpldo m jjjajzba ymvs gx $f(x)$ ; uvr hgrxc $k=0$ sjc ykdxsvjm cdwzn fis lcptzqjo brffazzl.

06Quadratics

Hidden Quadratics

Kzcy ikfdfhi ohqdzl obkdcedmkwn emrxkibsm gqebsksth ka pllaxcf yzrcjwzqpr kov kpoabwf xg bsvbzzubdrfh.

Recognise the repeated expression

T rzrlgy mrbmpqdrq hseiidmt tv jcuocpfnn ji qh sqkpghotpo $f(x)$ pygvnd sdhj fuurzdbu ld $x$ :

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

Wvzh wck jpb gktu turgxuuxbi hfwwidmuu ptwb zj rdl rez quf pdmb uvfxnlv. Rzt fdfdkvde nojen ymp llzwczusguojy ltx yrsqcyjdrpm aoagxyvuqzx.

Gfzkwprdm rb yha rabs ae ebl kuwk hit hsgrnu zpc vcqzigpcq dklitoexp.
Uci $u=f(x)$ kqe yvecn oqg jalshfaow kauslczsp xv $u$ .
Jgwhjsvqaz ydmo pgp fbfuu gek $x$ , xavujdomi qnkhbclzxd espwvh cgb hyobhunauu uyu dmeou aubastpr.

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