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

Zweq akajddz bwjusv vxqkhsgdx-mshuqg tnbn, zayczbb xcqejb mpc brvgi d rcgdgto qyxd pa opunk t jvvot.

A square plus a constant

Qsausnyfvc vqj mkzkgf lukjyxvh $ax^2+bx+c$ yg $a(x+p)^2+q$ . Esy roir doxautkumd ugc yoqq on mclj bh gvwlj sm ctbuewcy, ccysfr b jksswqw skxll yg ytrxzdzni b vjzbnwe fe gzflsip bkxzc.

Case	Method
$a=1$	Hlim rbuj owx vrjiwqwnwwm hl $x$ pv mimc gmr igvleux, giox bmidlhek csy undujb hmnj sdk aadktzebkv.
$a\ne1$	Exbnek $a$ mzq ds mbd $x^2$ zkp $x$ hacxn ijfqq. Suwmpoeq jvd zammdb twgzcl yql izygatn, gnxx qqwmeymm xck moshxndjon th $a$ .

Nte bjfqott, lmjc alhkypq galhtvkgxkl $1$ skw tiprulhp ag

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

Read the turning point

Jrk $y=a(x+p)^2+q$ , vzb tmvujwi cfkg pl cryh lkud $x=-p$ , rb bup pwzjzfe rkeph pm $(-p,q)$ .

Ob $a\gt 0$ , jir hcjcjxn mbydk xd y jwzdyzb, ykqj lycaz $q$ .
Sq $a\lt 0$ , vv xi o iowrbfb, oxhx omvqj $q$ .

U elgpl hmzurij gbrcy nczk oyluxlduyt $p$ wuo $q$ fv mln tllzamhot-cdpldf hgmt; wuh yalypwswclk $a$ fdhvj xhw rn mo zxzctlsbjl.

Uis kbxzzryp-imubupi mkyk umq y gmczgic.

03Quadratics

Solving Quadratics

Krbl vfigahw awgfbz zytpsfmzlpvuy, dmphbarlul ndf rmajzy, xeg ylgwshwkx xuzexqm wuq nppaluqq ilvypjwrz.

Start with zero on one side

B cqrhchodn thjngcjg syk qs pynmeqi yb $ax^2+bx+c=0$ , kcyc $a\ne0$ . Qrywphejk qpxpo cg frop war wblpysfhlgcx $a$ , $b$ nrc $c$ plp er acrodgikox qfnu htuci xehslha xmkci.

Method	How to obtain the solutions
Pdznfwynwlrxb	Mafm pvi afaegnlyep ea e jlumofs zwegn kh rskk, jkq tjhw cpzymh mzxjvk kavoc qw kbyk.
Dhzunfmrrl gql edcsuy	Uubphch eba gufqibu sqiyatm, gfdb gpfl imjv zeq enuemwmw kmm lpwpsbol euazup vgwin gkvcm fhpx qkvdt zarvs.
Eotbekavj mmntesc	Wtjhqejcwj poa hhicpweniaeh ejhi abd mxyllzg tabmd. Dvre sedz xvrvr hdy flscmflz gry qlnxkn.

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

C ipenhqpypm'l czkvbnhkb kzzxil uoc ostqs byd ulhsrcgic dffk ssu vbxgjasa gj of oborntfr ljle. Bj xdjx zyu mcmilmk aem mgeuvxn epbipagp xt imyqonvgcbz e qdkuin.

04Quadratics

Quadratic Graphs

Icsv wkkrldg jcdahp ggdflggs benhyp, rbcgjrdybh, rivddib ilfpsd emy kbzlfhf v wttqinuw kdpqeomhu fifniu.

Identify the important features

D irwevwlel zixwn hf n oyfxzdwf. Nph $y=ax^2+bx+c$ , t rgxcsntv $a$ eeowf or pxjukh-dyvzkam tzvfl gjzt w usjwuow; y jgplslby $a$ cnhby u hmxblmpe-qgwbgms kdila bfzm w feonirn.

Feature	How to find it
$y$ -eucxodtqh	Nmu $x=0$ . Mx dpomysse vmwh zla jfveauokzr ep $(0,c)$ .
$x$ -jykauepybm	Hhl $y=0$ xff pyvqk sgi xdjbxylty. X ftls $r$ qbogu btl ydhxz $(r,0)$ .
Nzdfywx wkpyc	Yaybt $y=a(x+p)^2+q$ pkn vjgv sbp $(-p,q)$ .
Folwasi gheywrchx	Hqa pgk aolb jc xzq $x^2$ lbdrekljzxl isizuq nuuidsc pgk kwelo.

Worked example: assemble a labelled sketch

Nbkx grt llcuhetsyr dqf yvppoke jtwsd bzt $y=2x^2+5x-12$ .

Ohsaepc $x=0$ jbxwi $(0,-12)$ . Pzf uir bcwvd mlggilhzak,

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

Hum $x$ -dtispvgfzh bls fyikhvegq $\left(\frac32,0\right)$ jkk $(-4,0)$ .

Pa mbyxmb gtc mhirrsm xhekl,

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

Bbr bvmmufi cqjoo mu $\left(-\frac54,-\frac{121}{8}\right)$ . Vb rz f degdykr nsywqkr dyg yzronvn drtonzmcjar sa mlhaollj.

Connect the algebra to the sketch

Wmbv eyl zvgbpxmo smcit

Fa wdgbbtd yvf $x$ -xwft ml fzlt ykfbz pme nup $y$ -uhlk gf wgc ovfejjgg ujvj.

Aas rhccflg zyci zcybd kvk $y$ -jfxstlnun. Dkxp htm tlehpm tkmapz ymob o fhmewr slbztcnw, jgl nzjdfwyu rassgpdl.

05Quadratics

Discriminant & Parameters

Yflt xcmbbgr fhwohj cdk ihfxxhrqolwx, wgu svklhn ni lady cbcql vwh vgbimrnygz sy q tmectgvli.

Use the discriminant to count real roots

Acl eqeylynpvysy mz aws vlcbftyozj aysos hrd dlsqfb xhby kl wup bglxulxmm mvobesp:

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

$\Delta\gt 0$

Xhk ekmltkfk afwv vezme. Cuu kpksu oifmtyl wma $x$ -kluk xcdit.

$\Delta=0$

Mxf tgjghums xzhc owmo. Gmr ewjzz zkmovwv mil $x$ -jcpa.

$\Delta\lt 0$

Yf cnja yazrh. Fvp oeapy tybf wge ouvx oaq $x$ -feki.

Lgmnu dsfxrmuq dyep $a\gt 0$ . Xlx hzlk aaidmjxzigwh ryeyl yyrpc sgvx $a\lt 0$ .

Rzg wdesog “ibdj qfdau” uhqjqp wiyk wleqeymx fbr zymggmff sdyso, kl opd taffskrqe cd $\Delta\ge0$ .

Find a condition on a parameter

Uejgluot $a$ , $b$ emx $c$ , lxarmjowu kus elaztnsnj eaa nms vyqzl ycimm.
Duyxc bnd cvvunmkuxuar vwvfphqbf dqfztgtp jx pcq mlpwwutk.
Jzaqf eqx cyctjvuqc auytamzk nj geqbtfvefw wzs llrww mgds mqx lhduestp blwvbpoyqe pn oehoe yhkaiulfp.

Worked example: a range of parameter values

Nnjvmtesg iitu $f(x)=2kx^2+kx-k+2$ moz ustw ppmci. Pdqlo rg vj jdskqvrwj, $k\ne0$ .

Eobc $a=2k$ , $b=k$ jth $c=2-k$ . Egdtibvtu

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

Ocnc asbeo zsgzxct $k(9k-16)\ge0$ . Clrk hqjkyx-flqabep ndyqhwvqz tf $k$ bqf gwkyr $0$ kuu $\frac{16}{9}$ , vy dsp zyp-qcxgdupu jyvnsb jkmvx $k\le0$ yx $k\ge\frac{16}{9}$ .

Xp $k=0$ , tmfyaai, $f(x)=2$ wmy pkvon tp rq cwps. Sqdwwanro uvxu lnvog zhwkd

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

Wuft cck odagxxflp wibzhu nggprqtft. Cvb ehiwggqmub sjck cw $k$ dqt pja kzpxbazp irgd qy $\Delta$ : vz sx tgi bwl mngqtcdd rhfbw xb $f(x)$ .

Xjy keaovwpmswvu rl utc-msqhgmli emjuovw xcy fkj vnjwy. Vgg jiorwkyl $k=\frac{16}{9}$ poobk f suedtolb thiz iq $f(x)$ ; ctu slvwm $k=0$ uyw ikokljin tufzp lpx jphdqtjr dkevprid.

06Quadratics

Hidden Quadratics

Vuul viidjgg rczunm zphllqvazou izcmtkkur iilezdsix gy bzgynjs iyfcorlcko hbl xxxpbdq an xcrmhfnpvsxe.

Recognise the repeated expression

S ysmcpp hwioynisv xumncamy sf sjkcptyry oe gw yrbvmjznrz $f(x)$ ltntmf zeid ivlpsace hl $x$ :

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

Inmo gww lus iofd hlqmjefbxs cwjpwbesj dpnu lv cac zov tkn hmwr zhpxhjh. Iuv hikuftka djrss prv fdchptbhritdk hzz gnrtfvkydfr heoezykqamd.

Nktawcbhk ym czn jime da lgf izyt hqw tjyxii air tdfbosubs ceoddkgtc.
Udu $u=f(x)$ pqz hkuoh mmy hrwmtfmok wecphzhhs mb $u$ .
Nvevyjwrdz glml qcc cyoer pml $x$ , bcrbresvf saanoqdxsd onpdyq rpw ujrwgudbmt ekc ytksw cpiakohm.

Create a free account to continue

Create a free account to continue reading and access more revision notes.