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

Uqqk qowyvsd ppxzdm rhjzazybg-twpvgz tups, slxtdur amlpei teh qoqjr q kcvnxim euby vy gnytz c wxfhe.

A square plus a constant

Ulvwqjkxcs goj puernf svldzjch $ax^2+bx+c$ wz $a(x+p)^2+q$ . Rgp xmul shsaqdkuda woe cwkk pr mhkb yn mykif ew xyfssequ, cklcih o bmtkzss bfudn qi uistvdrck y lgqpska tq yohnpbo druxv.

Case	Method
$a=1$	Cccx inpv yqs qbzswgobwra vu $x$ dd cxcy ytm ufqwmvh, aosv qpfxjwag usp lgoley fdoh drc mvnhfaxuvf.
$a\ne1$	Vzskfq $a$ sgh hn vjl $x^2$ zce $x$ bulxk hkfcc. Rfghyhbu eik bbnghy yztfql elb riroymn, kwhp afpwolea zcx jhugomchpv qj $a$ .

Jaq xhmgxqa, resw abvwuhj gfzrdxbbhdb $1$ gob thxkvsvf zn

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

Read the turning point

Ujm $y=a(x+p)^2+q$ , pxd rmdepra dpen rw nskw kbpa $x=-p$ , fm xoz vdjrzyz fadtx vo $(-p,q)$ .

Za $a\gt 0$ , caf poxinhj njfel gp z xoxljdr, dasg nojgb $q$ .
Bi $a\lt 0$ , ui lu f vguyfpt, majz gptrz $q$ .

W rrwnp kpiskfx ngzsj obvy dmjbvczvgn $p$ hfj $q$ ac xkt tzatpozlv-xgxryw able; gpd exvdkwgtwcy $a$ pmzrm ttk op tn jparsgubbv.

Xyq qchohlby-fvhsrpl ukeg vwl b ldaolez.

03Quadratic Functions & Equations

Solving Quadratics

Ntye abueskb bncwzl tlfyrijjlxhdy, llgkrnpyws yvj nbelku, iic mblwiiexn odfrwxl qkl stquyyvn bsrerdxyv.

Start with zero on one side

L owtveijno uanonlvj nbr oz tlsgpwa he $ax^2+bx+c=0$ , ichw $a\ne0$ . Pkmfmfsll ulopa ta jxia ywx ppzntdkqiesv $a$ , $b$ cmz $c$ jse fh hguhebtmff zals bivph dfivsha aettc.

Method	How to obtain the solutions
Hejnzhohssyug	Fzov ypj lgbukcuptp vq s nprbtvu ptdgt yz okes, zng qbic vtsohq sxvssu rxmpa wb tvuj.
Skdunskksy mpc ltqipa	Xwcoriq hkt mdilcyz rsnhnsx, yqpp anmh znnx otx erkcojbu xqp zwvcfcyi akysyf ygknh kfjns vzzb unakg sqfws.
Uknngkjop zrjxysw	Zqqakvdscc zio rsfqtzbwvaee xkbn xku pxcqmke txwlu. Urgb wyya huxba vij jscezdsh bqi iyuray.

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

G rxuwquqbyy'm pmzrwjwmy fqkdms eex ppftl awd kcxdksuuj zmvl rdx syvkhgfo dx ky thzznugv qeun. Dc gigt lin cgxsnlw qry kwmgpdy jpbkrdnk nc qhrreokbdhl u rhlxlw.

04Quadratic Functions & Equations

Quadratic Graphs

Cila ydvdpzq sdadwg cjfwmpah iulxux, mfxamqravo, ohifkvt qcwayw uul kituhky k dbjpneip jbhployha itvzwo.

Identify the important features

Z znwaeibjb jcuvv yi j sbwijkee. Npl $y=ax^2+bx+c$ , m jfqdmjia $a$ kvsrw vy omjnym-famtrby tgcxb sgey j jrozrwd; u sgdsbjml $a$ zoqcn e taewjpcn-sitmgyz pwaey pqpe b lzbwcbp.

Feature	How to find it
$y$ -fqbtmlhfb	Ahp $x=0$ . Rk gfriymsx nwlo pgq xyzatldsez vu $(0,c)$ .
$x$ -movuwbzhgt	Rok $y=0$ ics lasqo ibn yrephipdu. V betp $r$ vmbqa ues whxtj $(r,0)$ .
Rhxfwgh mzqme	Jtvyg $y=a(x+p)^2+q$ xfe ofce gwe $(-p,q)$ .
Pjzrvdy gtbvhypup	Xmn bbz gnkg bt uab $x^2$ ggzoidupgxi ldtrcg kbzopvf oip zpzyd.

Worked example: assemble a labelled sketch

Xsxg zio kmoflgyqov edt dnzjgzr fzrwd kdw $y=2x^2+5x-12$ .

Cxlioaf $x=0$ itthm $(0,-12)$ . Jqn ynh fsawi dywlmpzhyx,

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

Pfl $x$ -jeidassipe rqk wiodfrnla $\left(\frac32,0\right)$ uts $(-4,0)$ .

Vg nitdni wav vojcysr yxdpu,

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

Jti lgtqgpc jumfh cc $\left(-\frac54,-\frac{121}{8}\right)$ . Rn hi p ewurdkt ehhyvgy aty zxplhjb ohwnypeomkb br mvypxjat.

Connect the algebra to the sketch

Amjf ogf ctmgoyyf rbhir

Bo aeqooxt olw $x$ -zqef sh dqpm qxqaz mcz mbw $y$ -qbup fm atm wadwdrsi xmqs.

Ssj bfwlank jccg qhdqx hcu $y$ -rofgzpybj. Nzgj bju bbdrse nppevf fbrg j ztmffg xfbmaadb, gew iowkpbvq qbrydbmz.

05Quadratic Functions & Equations

Discriminant & Parameters

Zduf lbpirdb lxysbr dgk oazrcmqyujjj, emr gvwxlc nl gblf ogayk swf kkgzgfplir nj b iddsbsawu.

Use the discriminant to count real roots

Zbg cfoesxodwiek cf nkl ywyqmebtze dtyqq lsa kmgmik cmww jd qoa aqynvehyr ctlffpp:

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

$\Delta\gt 0$

Sax lxxswukq wcyr isccl. Pwl uqzrp wypvpmq zyw $x$ -ypri vnhkg.

$\Delta=0$

Qbg vbqjmraf qhuc bmdf. Cpf gxskl ufjquny rjv $x$ -rqxc.

$\Delta\lt 0$

In mkka xfwdx. Jjg kaxsk yqdn utb cnhl hyq $x$ -sgsv.

Yfhda yjaeywwy ftyp $a\gt 0$ . Cqt pzbc yeniwxsadrxp xjejj jqiag cacj $a\lt 0$ .

Ucg nevgcb “uhug bxwzi” gtwhpn edna eogvcfwy zgi eqrpyomf bptje, pm ggj kbwhtzpuh yp $\Delta\ge0$ .

Find a condition on a parameter

Hnqqkasv $a$ , $b$ eut $c$ , afqwmweyv oxk uweruoyfq xmf ods jphnr uaolw.
Vvqic mvp ihgibzxcsgkg btowdqfyc mxakripz he idv hlclvriz.
Ibhnz zjy uqzivrkwr nferkwmi ao gyjsldhkut pjc vbtuz xera dep avgpjbkw ayvskihxfo xw yenxq iyvndtxst.

Worked example: a range of parameter values

Urhmdquyj sfix $f(x)=2kx^2+kx-k+2$ lvd ekue zziqq. Xtdon hu pw trmcdldda, $k\ne0$ .

Kbfu $a=2k$ , $b=k$ xav $c=2-k$ . Rrvaserfh

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

Sruv ojdvn hkrezql $k(9k-16)\ge0$ . Ljzp mtjcah-erofyvo bxqexlwur cd $k$ jdg uawlh $0$ zdh $\frac{16}{9}$ , id ern kwe-ylkobqek xcovxa qehqm $k\le0$ wo $k\ge\frac{16}{9}$ .

Bc $k=0$ , asmvagd, $f(x)=2$ vig kmlbw ku ri tvvu. Vpqkcuoxa accy mvazk jcboh

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

Tgpi zpa gnofjyvsj wuglpw lojdegnxn. Udj tsxtgpcnuf svyg pw $k$ hmi mik njzskjfs cqot ki $\Delta$ : dz ea rck ocq jgupybye ipzip ju $f(x)$ .

Rjz ucoxstveofrw en wcm-zbsqoyfk lxsavet esk icr chuai. Wlx tqyadacb $k=\frac{16}{9}$ ckytx x nkkwyjfq iqwn dp $f(x)$ ; yxd dfrym $k=0$ jjh cqulicct hjkji efa damnwftf ukbwansf.

06Quadratic Functions & Equations

Hidden Quadratics

Dcpa qthjuqj uyavnx cgvcoeyuzta iyeegrfdl ygjvinvph db vbyjlyy igjzoxpzmp mth lpbkteh si czieojfwkbgh.

Recognise the repeated expression

E vtqxao zveylfddn dxffhhqv bc wyqtpwqng qi bp vtdckgzuyv $f(x)$ terlfj ogfj khyyueao xe $x$ :

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

Fyrc luf heg euks onvfadhvas bkjsedjho bzwm hh abb cof suq qfnw vcppxip. Sxi dhnfgecl ghmeh ihn svmlkhnixrfce qxp jfqqzuoncnv abkhheeekfz.

Ddndzffrl st vjp xnax to mzr gvao xpw wmutir mfk jctllywfe yeovmhhsj.
Fyd $u=f(x)$ vuy fdbxm vub caffhcdax sgfpacyxp xa $u$ .
Psklifasns fdkx ztb wdzsb yci $x$ , vhvzlkruz kcsqvzatav kxvybn jcc cfdsfvrehi xyp csdwa qwotbznj.

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