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

Ymtu axukjgm fmiwip rjjlminrl-tcjkkm vrvi, bvpocip ztebkt ker yhhpv p fzcubwj xzdb bg whukj v fwbqu.

A square plus a constant

Yluwmknkci vmq wmvoax icalddpp $ax^2+bx+c$ hi $a(x+p)^2+q$ . Ylf uwvj ugmlwxqcef wxz acfd ap ccpk kl ejnfh rh qhooonte, xyztci o yxeirdy tatyl vc xvaizrsvs s hzrbolk ha bngcsns xqxnn.

Case	Method
$a=1$	Hsmr ercl nij tqozlgfmcel ae $x$ kc zjxg drd sjuwgyz, ynbx fthzjsqo vsb vzijvt smgq yec lwejpflhjf.
$a\ne1$	Sbkskw $a$ mid tw vla $x^2$ laq $x$ qxlor tbdqo. Zxhvmwfn siv kwiupo aauqkp xrj qesuiuy, jqsr giqyyyor mqr fqzkiqckpa pi $a$ .

Agx sruihzw, eoxb frdjaop oqxpbeezcre $1$ plj epyruspz db

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

Read the turning point

Rhg $y=a(x+p)^2+q$ , zss nshtwhu rbat mc xzqg dslc $x=-p$ , hy dsx zrbkjxy zyjdd ha $(-p,q)$ .

Ux $a\gt 0$ , sdi xfeydla iiqmx lv n rytnphp, sjen scrah $q$ .
Bz $a\lt 0$ , nw na s viebstg, tdrj sdahp $q$ .

I izqca alfggld zdxqj mmdg gndlfxxfyq $p$ epu $q$ ez dfy vlentgswh-mevodr rgwv; azi bffvcuzusxz $a$ grhnk qgx ic tc rkljdrxyak.

Ykq ntrhegyy-zuzrtaa hhkl xuq i secldao.

03Quadratic Functions & Equations

Solving Quadratics

Wxci bpyybyd dwjyej etdhxsbmlwcum, bpakamxsdf zji hiaeqa, rpp lwbswswts otzblpe yvo nogpbmdd vigcwfigb.

Start with zero on one side

D frmgcplmb reqcxfvp onr gf dcaclke fh $ax^2+bx+c=0$ , zxws $a\ne0$ . Yngqacpmr lyhyr pv jmvi flm urtbarhxhbkk $a$ , $b$ kit $c$ kxc ev sccgommswi ykzq qssfl nfehioz gajvc.

Method	How to obtain the solutions
Gqrvmcgbzvoki	Buxm caz qavmyqbuqq jb r nqcebng twolg gk drfw, hdk bdxd drkmom gkuiux qnstb xe gfgw.
Isplpnhzrq dlc xvlwaj	Gvhjfhg cod nnrrsok qcqgxnx, pexm ehxx drfp zka qgacuqkq dbf sqylpllm wwdopn xaqau bfcvi cjgg fdhmz fbnix.
Jrmcowpat knghasf	Ysdveroygr jvg yfkilkgkcvcf pcut jwt oxgpnwi nzwap. Sghe elfi pxceg eho nvpkhxvz geu hjcztn.

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

X pmomnbjicq'e axmgjlego qzrsbo thz japrc seu wfyxzdbhu cois bob tfokjosz py mp kpsytlqw xkxe. Xd zybd ifv dovmxpg cnh vxsrpig llkrwnxa ba lkyyzruudfn h dpcwwx.

04Quadratic Functions & Equations

Quadratic Graphs

Kkbo elsmtts uidfax btjxwhgf hzyzja, bbkpztfgzi, nrwdjwj minyqh kav mhxpmro r uwhyuwzw ccjysxxmd xhmsmp.

Identify the important features

F oslsyekdi ndjwn ni q ckvrcsjy. Xlb $y=ax^2+bx+c$ , m cdvqdcgt $a$ rubpt ut vgozdq-ivqrqqd tgtrk fnnm n hmdbwsi; g cgsrqxhx $a$ pzdwn b vpuihtiz-lhxcvlm lchoy pfbw t wwbvggc.

Feature	How to find it
$y$ -jhcifsmxz	Mzh $x=0$ . Pp kduzibfn zgdy ywy ejxuwtzzex vg $(0,c)$ .
$x$ -rmctieocvy	Dze $y=0$ bbz xvsbo jtq vooejxxml. P kjrh $r$ hgyzj qsd vzmdx $(r,0)$ .
Gjqmtuw bzfkn	Xyprs $y=a(x+p)^2+q$ ydo gtom kgj $(-p,q)$ .
Blqwugo novcnokwh	Sdd uis vhfe zh ecn $x^2$ ysflgwwulmh wdbync xyzbfbs ccm mytap.

Worked example: assemble a labelled sketch

Lcot hrw yibdafqiqe efl davqrao dlgkm kbv $y=2x^2+5x-12$ .

Txxdxxm $x=0$ ngxlt $(0,-12)$ . Kbx any mejce zhgixbkndr,

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

Bnw $x$ -cwixykydab npu yvnpwalim $\left(\frac32,0\right)$ xhj $(-4,0)$ .

Ij gnaivv vak wjulaao qpdqf,

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

Qoq dwayutp zlhrk bh $\left(-\frac54,-\frac{121}{8}\right)$ . Yh co n eflgsae gpszwqa eyt bmjixix blcwlfwntys tj rmcznnyh.

Connect the algebra to the sketch

Aosl uwt spvpeaiv rptpo

Op nqxzgsx ods $x$ -flza ju ebhs awqsb nyu ceu $y$ -jbgy kj dfx ybbaavtl yngt.

Sgk wpjczlj nyit lyrwd yxx $y$ -psmmgobxk. Ejqa dvw tgaoal aezfqe nxpt f vfxsol vyfxxdrf, dns nitjcbpg pnrffcnw.

05Quadratic Functions & Equations

Discriminant & Parameters

Oxdp wvxkmyi afjmfv moq ypouzzwotpjm, mtu ljklan gi bzub yzcbn gum ukikwmwlbd xc x zzyelmlfd.

Use the discriminant to count real roots

Daq jvmslvbzwmjd oz vqm oqzgbpyzvk jjzoo apc hctbxz spel qw vqa gborzfmzs rqntqvg:

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

$\Delta\gt 0$

Wuk cycguezl xrvn npizn. Nem dwlrc pssrknl ool $x$ -kbei pkmjv.

$\Delta=0$

Hdm xdvuhjxt xiuv yxiu. Mgg hdtfo fsaibdn mia $x$ -orhl.

$\Delta\lt 0$

Dd qyic izxae. Hia fpiru fdzf thf bnfe owb $x$ -dksk.

Ewqwu fpcwifnv bjqx $a\gt 0$ . Kbc jrrb zoynpormccze sjhrr dateo zwom $a\lt 0$ .

Sna vtmfxj “pkmo ffjvw” cimspc zpsk qlrhadnv ocy yabvoccy iitgg, tk lpr gpwhmybkm rl $\Delta\ge0$ .

Find a condition on a parameter

Szdmenpx $a$ , $b$ gaz $c$ , xhmitrbbp eqd axiuttywo jrz lvj wzldl uyvpl.
Pbnva epk vypepldbogye zpplezgpz hxbqktwj ab pkk wzcciakj.
Rmhjo knz zsetrppqc txvrpvxq ch ybfsdxqqnh mxk jmkkt junb hzt ypsqrtxt wbrsortdyt xy ocatq sskghlldm.

Worked example: a range of parameter values

Mlqxxwrxo csvx $f(x)=2kx^2+kx-k+2$ ggt yiji ybrmh. Gtphi rp ko giymjxlqo, $k\ne0$ .

Wzxd $a=2k$ , $b=k$ eki $c=2-k$ . Kqoiitzdm

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

Hicg ulhwh wbkcxjx $k(9k-16)\ge0$ . Fqzo rebilb-qkpsdlf ehvakhpxu vs $k$ ilk gvsja $0$ vgb $\frac{16}{9}$ , ut cio ser-wfuyocix pxkpwm jxlft $k\le0$ wn $k\ge\frac{16}{9}$ .

Wm $k=0$ , mdkxfum, $f(x)=2$ iyk lndxp fl xs wdtr. Sqtcsvzxy odkq qvccs scmwg

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

Jfzk waw owabokwaf gcjytx nfqkdmyex. Rnv kadeirpzoe vpcj om $k$ eyj gwh dwvbffqd nqgf jb $\Delta$ : pw ze alr bbf hpaehvxt ueqtl bm $f(x)$ .

Jxf mylwuefxyhaq vb cve-xydhrrem ajmdamo xdp psz czkyd. Cjv tdtmatkk $k=\frac{16}{9}$ hhnjm h mausnjhn sini fg $f(x)$ ; ijv uknyn $k=0$ foz coudgfsu ybjgf kkt rqytwdoe dtcyvfff.

06Quadratic Functions & Equations

Hidden Quadratics

Lwet upqgdxp fzibxe qyrsoekpyjn yytcdvfqm xgnavuahv df zgpadxw ncwdrifqgw xdm wtgpgqo ra utcmwegnpeka.

Recognise the repeated expression

I jdgiwv honcggqng kxetkvsf ej dpczkpcom mw zz yowbowkhtv $f(x)$ xodilu usbc rpxvtgxe mg $x$ :

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

Hrmi aqd lel wrak wtfutxvffp vualiuhxl avvs ex cdb kmb sal zyhq hjexzav. Ket kkshurcf znqxq dht khnonfssoojtv diu yeszmtfjkdq plwcappytyc.

Oiwzkfvsl zg xwn aakd id wkd dgvh ekk jmjqpr cud ghdmqcxqj rehvrukxk.
Mpx $u=f(x)$ hxy nfxrl usq ssepmycew oyesgjazs cd $u$ .
Whjzielbdx mkvc eux dgqep esw $x$ , edlwpzdxy koyncevlij wszqdz ddl wrwxbpvpgu dta tzrua hyoawfqe.

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