10.3 The Quadratic Formula
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1 . Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti Fomul Te solutions to te qudti eqution 0 e given y. Sine we got tis fomul y ompleting te sque, nd ompleting te sque lwys woks, te qudti fomul will lwys wok. lso, sine it is mu esie to use tn ompleting te sque (s we will see lte), solving qudti fomuls wit te qudti fomul is te pefeed metod. Tee is one speil pt of te qudti fomul tt n elp us detemine te type of solutions we ve. It s te epession. We ll it te disiminnt. Te Disiminnt. If. If. If > 0, te eqution s two distint el nume solutions. < 0, te eqution s two distint omple nume solutions. = 0, te eqution s one el solution given y. Tis seems fily le if we just notie tt te disiminnt is te epession unde te dil. Wen te vlue unde dil is positive we get el numes nd wen its negtive we get
2 omple numes. Tus tt teizes te solutions. Fo te lst se, if te vlue unde te dil is zeo, sine 0 0 nd 0 0 we lose te wole k pt of te qudti fomul nd just get left wit te vlue wi is one el nume vlue. Emple : Detemine te type of solutions te equtions ve Solution:. In ode to detemine te type of solutions to te eqution we must evlute te disiminnt. So to do tt we must fist detemine te vlues of, nd. Sine ou eqution is in stndd fom (eveyting on one side, zeo on te ote side) we n simply ed tese vlues off tey e lwys in ode wen te eqution is in stndd fom. Sine ou eqution is 0, we n see, nd. So we evlute te disiminnt s follows 8 Sine te vlue is positive, we know tt te eqution must ve two el nume solutions.. gin, te eqution is ledy in stndd fom. So, nd. So evluting te disiminnt we ve 6 Sine te vlue is negtive, we know tt te eqution must ve two omple nume solutions.. Finlly, we see, nd. So Sine te vlue is zeo, we know tt te eqution must ve one el nume solution. Futemoe, we know tt solution is. Lets do some solving wit te qudti fomul. Emple : Solve te following using te qudti fomul.. 0. u Solution: u. y y 6y
3 To solve using te qudti fomul we fist need te eqution in stndd fom. Ten, simil to te disiminnt emples, we n simply ed te, nd vlues off of te eqution sine te ome in ode.. Notie tt tis eqution is ledy in stndd fom. So sine ou eqution is 0 we n see lely tt, nd. We now simply need to inset te vlues into te qudti fomul nd simplify te epession s mu s possile. We get So ou solution set is i i 7 7, i i.. Fist we will put ou eqution into stndd fom y moving te ove. We get u u 0 Now we n see, nd. lugging tese vlues into te qudti fomul gives us u
4 So ou solution set is 7, 7 7 Notie on tis polem tt te vile fo wi we wee solving ws u, teefoe, wen using te qudti fomul we d to use u insted of impotnt one we will see in te emple... Finlly, we stt y getting te eqution into stndd fom s follows y y 0 So we ve, nd So ou solution set is y y y y y 6y 6y. It s sutle diffeene ut n. lugging into te qudti fomul gives 0,. Finlly, we n lso use te qudti fomul to solve qudti equtions tt ve moe tn one vile involved (lled Litel Equtions). To do tis we just tet ll ote viles s if tey we just numes nd solve using wieve tenique seems fitting. If we deide on using te qudti fomul we must e vey eful out ow we oose te vlues of, nd. Emple : Solve te equtions fo te speified vile.. fo. fo
5 Solution:. Fo tis eqution, sine we e solving fo it seems tt etting oots would e te most effiient metod. So we will poeed s follows So.. Tis time, te most diet metod is te qudti fomul. So we stt y getting te eqution into stndd fom. Tt is 0 Te vile we e tying to solve fo is. So we need to tet eveyting else like onstnt. So it migt e useful fo us to visulize te eqution s 0 Tis wy we n see tt, nd. So into te qudti fomul we get 8 So,. In tis emple we n see te impotne of oetly leling te vile fo te qudti fomul sine tee e nume of viles involved in te polem.
6 . Eeises Detemine te type of solutions te e of te following equtions s Solve te following using te qudti fomul u u y 6 y v v v t t t 9. p p p 0. y y Solve y ny ppopite metod y u 0.0u. y z z v y y y 8 y Solve fo te indited vile.. 6s fo s fo gt d fo 8. s v0t fo t k k N fo k 60. s fo N n n fo n 6. 0 fo T T fo T 6. V R fo 6. y k fo 66. d y k fo y 67. y k fo 68. y y fo
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