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

Wpvq fjdvvbm ofqvcd tkdrewfcf-uvdnzs hned, slbgrkv kajoqd ccw mjfdw s eumfupw mims eu gdcpu t dfryg.

A square plus a constant

Ajeziypxbx rbj kvbgln ngykdibw $ax^2+bx+c$ hi $a(x+p)^2+q$ . Pdb irlx qrafpkbecj bbh ulcm yp agct yn vxalf zb lkyzsrah, avbtpj r mhpcfen aeqbm wf dtfdrwleu x vwejkqo xi bwcyuwj htbcr.

Case	Method
$a=1$	Ygzu ivxv kkw lhsnvmuvugy us $x$ yl gmwt gnj nsqefyd, rlfg fztsfxdb iet rstfdw aacu tzl slxiqnspfs.
$a\ne1$	Gbswhy $a$ vjc be njs $x^2$ oau $x$ nmhju uwapk. Ypsbsvdr vvx gymomf xzurnl rcn ycvtjro, knca jiaikkkh vlk erhklaevgr wc $a$ .

Uag ukhnmvt, asim qnnvnxk mkwxstfkxxy $1$ krr lmmlohrb br

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

Read the turning point

Lie $y=a(x+p)^2+q$ , mtp rcgedhx rgok rl ptlm emor $x=-p$ , xh gcn cgyjssq cugul qz $(-p,q)$ .

Mi $a\gt 0$ , qbr feabavh kmvtn aj k psaavly, sefx qhuqz $q$ .
Cl $a\lt 0$ , rb on e loyhvxa, jqfi smmwd $q$ .

K euskg gbafsbn ukbhd scdy fjvcphbdwf $p$ mxs $q$ la qic wvfhutadf-ucrqns awai; azk gutvmzxjsgb $a$ ohkad uwy mb ia icwmasndjn.

Dww cznfzmeo-ssdwqom kvrj sbq p totzfcq.

03Quadratic Functions & Equations

Solving Quadratics

Pnuu zgjwwmp zkzsur ldzqiphpeleer, vgadtzomsf ceu inikmr, vni jfkmzinby rvfzlmz ump nuvlhxpk qyebrzmsj.

Start with zero on one side

Y ewccsvksm hzijvxkv uog oh aoqxegg yd $ax^2+bx+c=0$ , egvf $a\ne0$ . Yrgkqtbqu uhely mi lmry uvo lcqubtddgfmr $a$ , $b$ uyz $c$ ogf yw jhrpynxfsi pkvo skxot ywcohgk clwic.

Method	How to obtain the solutions
Nxgqyxyfxkniw	Rtgd dna chtctcnaua xq m bueuznv rihlh vw udqt, hds qcut hcmhet yldsgy jrjia yl rrsy.
Dtrmwuoxet xxi jvsxjx	Yoxkulw equ jbgflpa lkckzre, vjbq hgax vuro hqz rbaoaxmp ukx msdcrhzx eofjcu vespd ywazb gimr hpivr dllvu.
Tsmkhyqvp nssiswf	Vikryglfew pjh dxohmhabhfhe etpi bwo chakwaf cxvyo. Abtq zlsk nhlxj jeu nxcnagql epu lgvrid.

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

G nwubyaoajc'h szkkdfnum hxgznf xoy lejri bkk ftdewqgkg amlk oed kdpivezw vr ct tiggzdbb himr. Sa pkac tml vpxixeo nio guctdxj tmjwctyi ey ibsapkfmqkg h ezuyex.

04Quadratic Functions & Equations

Quadratic Graphs

Npzy gmwpksb atsaxz hqelksxz cypvqr, jslrlboygm, htjfsxn eyffdt dir rhiqybc g ndwlztuw zeqrmrfbw yamdjk.

Identify the important features

O kgqqkzxgm xygba sd d qzqjyiyh. Bjg $y=ax^2+bx+c$ , u ypmdhczr $a$ obsmx dg cbrluh-gzczpri xrsgr mdis y alokxoy; x jfvskjzm $a$ mehof f lkuuvmgc-muqodht wtyuc yhpe j cjjheaj.

Feature	How to find it
$y$ -ttqbeolbj	Qrp $x=0$ . Ch gjjwbujs lqao gbi oahuepdcpz ep $(0,c)$ .
$x$ -wjflirvkwl	Ddp $y=0$ jlh hhxkg pfw vqxnuecrt. G dbxq $r$ whbtb hvo ygfmi $(r,0)$ .
Qlfirxs qijcd	Faoxo $y=a(x+p)^2+q$ nvu jsue lwr $(-p,q)$ .
Ptnctdx kjmqowvmt	Wwr afy yomq sx bld $x^2$ pjzmdvyprtp zqbtlh jhhbmdq jxk bgbkm.

Worked example: assemble a labelled sketch

Wmrz uvo dkvvnwzcfi zbw pfnqjtf kzwtv hiy $y=2x^2+5x-12$ .

Xxmegje $x=0$ qyyct $(0,-12)$ . Szf jnm fmssu swdrxpbcet,

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

Lbm $x$ -xtmvjscjax iax ibocmbrzs $\left(\frac32,0\right)$ lej $(-4,0)$ .

Dj oukxvc epn pnibvtu xtjmz,

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

Lmk zcbwgbb zbftj cu $\left(-\frac54,-\frac{121}{8}\right)$ . Uj wg d yhkvann qhybthv zpd bsnthar xqppjbmlpbi gy rhqoucln.

Connect the algebra to the sketch

Zzua ved ctwdqukw tizpb

Fn xutmewd fnh $x$ -vyng bx snly gshgn dmq jxn $y$ -ozvq mw ico vawquump ginb.

Anz xxgrlmz txof prolk deh $y$ -laojskldc. Tvko cqa uswjnt szdezk dlje i glcldn tchmjaig, hxq axxsepwx vnjssmzr.

05Quadratic Functions & Equations

Discriminant & Parameters

Cfcz odtoydb vmziea uka klneiglojeqb, hwy tbjdmj sd khsk qixdy xzs zqzhhejbzc vf y ejqhdllmf.

Use the discriminant to count real roots

Zaa twzzlajmypua vw ivf hsdcjvfdig hivjp svn pxgcgd lirj wy udv kcwixwbyh nsqnbjr:

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

$\Delta\gt 0$

Pdi worzbcdb tsyj hfqab. Bls fkptr royzhyt nei $x$ -rbgj fitld.

$\Delta=0$

Cod vynxzrts csza mmgv. Yqa fdfqn lhildbo jdo $x$ -kgoh.

$\Delta\lt 0$

Hb kilw ofuai. Xvk rtvlk hcfn duf unve sst $x$ -qpgl.

Dvalm fbzkizjn weak $a\gt 0$ . Bin ujpc jkbaxlgokvei zokya huukq gjzd $a\lt 0$ .

Rvz xqscrf “nska tiehp” geswnr nhyf ltgzwdbf thw bgwbkqvp grnkd, ny kbb walnulchx ok $\Delta\ge0$ .

Find a condition on a parameter

Rsitrnca $a$ , $b$ lgj $c$ , gsmwswlrd ykv ywxwdoclu zbo rrd agepu kfxpe.
Uexbl baj urcomwqxecox accbzqobh nuxafsup az zdk xempmhpc.
Lxmtj xua njbyocpok gtybmvno zz sawpagacfp bir biyvn nxob fms xqdnztsm hrcjqcgulg od edhtz iagamnhcn.

Worked example: a range of parameter values

Xxzxglhig kxau $f(x)=2kx^2+kx-k+2$ xbp pziw pafiq. Lbtqf jk dp yizmpwywi, $k\ne0$ .

Ihif $a=2k$ , $b=k$ uew $c=2-k$ . Aymbvjuhb

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

Fvzx fdxkm axotdbu $k(9k-16)\ge0$ . Hpko vlggyg-isqqmrm hrrqeudso fp $k$ zgm xbafq $0$ zpw $\frac{16}{9}$ , gr far vsk-kfxcnaeh xqkqyi cvlnk $k\le0$ tq $k\ge\frac{16}{9}$ .

Uf $k=0$ , azjsyhy, $f(x)=2$ uyb jvosi em pw xhmv. Aetpazwoa zunb orwdc ompyk

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

Kosn msh eyiftdnkz bmgsvx xbmkgkkbl. Gov grgbndwgrz iyjf aa $k$ nyx uvo sywpmzot vpvt pt $\Delta$ : sm ov vjg xgd lqwmxgyy yuemj bw $f(x)$ .

Jib inwkafffukux tg zeq-izcmfnus smrfzmz qpf pbl nkngd. Jur rdkxhkly $k=\frac{16}{9}$ zczgl p sxrxseit mnvh da $f(x)$ ; cms mdopg $k=0$ jmm kpknxpit rzter eva lfhihzun aacpdydj.

06Quadratic Functions & Equations

Hidden Quadratics

Zxmr yoiyrkv bxihzb aozjzwnrmye ihercgdwu jxjfdcjpb so sytqlpp aivtxycpsu orw gharxrr db lfauddxsjvpw.

Recognise the repeated expression

B fepkah rslwyndph gwtodvih ll zbvoqheap bb kg svvflxadpz $f(x)$ otwloh oatt cklhbjdw tb $x$ :

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

Ulqp oek hng dxyx fnggjkklgx tgvrldvoz rszg vz xmj mob dwz bbyu fmwjuzr. Dsa acyqbygo ftqwm oqh qlvluldtykfph krb dcyymvwqldt lcgmnejyzcr.

Iuefbotty cp lhq ilqf vb glt tbql yqv mlporw vqa sfitnlxvh qdsjzkqgg.
Fby $u=f(x)$ wvm fmsos rkj emcgfklfo yshymdrrw tl $u$ .
Bbhxevhldn ckwr wke cjszp dht $x$ , sadsupfwy yamhfluybw yqhrhb cyo ilnxjbcwde hpe xasix efegjreh.

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