THE DISCRIMINANT & ITS APPLICATIONS


 Ami Dawson
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1 THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used to provide n indiction of the number nd the type of solutions (roots) tht will be obtined when qudrtic eqution is solved. If Δ >, the qudrtic epression hs two rel solutions, roots or Xintercepts. If Δ is perfect squre the roots re rtionl. If Δ <, the qudrtic epression hs no rel solutions, roots or Xintercepts, s the squre root of negtive number is not defined in the Rel Number System. If Δ, the qudrtic epression hs one rel solution or root or Xintercept. The curve will touch the X is t the corresponding vlue of. The discriminnt cn be pplied to combintions of functions eg. A line nd curve. Just s long s qudrtic epression is obtined when ll terms re bought to one side of the eqution  the discriminnt cn be used to determine the number of solutions. For emple: If the resultnt epression is not qudrtic function  use technology or other lgebric techniques to find the number of solutions. The discriminnt my be used to find/prove the number of solutions for ny theme, s long s the eqution describing the theme of interest is qudrtic function (or disguised qudrtic). For emple: If the eqution y + models sttionry points then the discriminnt cn be used to determine the number of sttionry points tht eist. If the eqution y + describes the points of intersection of curves then the discriminnt cn be used to determine the number of points of intersection tht eist. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
2 APPLICATIONS INVOLVING X INTERCEPTS/ROOTS To prove tht there re two roots or two rel solutions, show tht b c >. To prove tht there is only one root, distinct solution or one solution, show tht b c. To prove tht there re no roots or solutions, show tht b c <. To prove tht the roots or solutions re rel, show tht b c. To prove tht the roots or solutions re rtionl, show tht b c is equl to perfect squre. For emple:, APPLICATIONS INVOLVING CURVE FEATURES To show/prove tht qudrtic function is positive or negtive for ll vlues of  show tht the curve does not cross the X is. i.e. tht there re no X intercepts i.e. Δ <. i.e. Show tht Δ <. APPLICATIONS INVOLVING POINTS OF INTERSECTION The discriminnt my be used to determine the number of times tht first nd second order or two second order equtions intersect. When Δ > there re points of intersection. When Δ there is point of intersection. When Δ < there re no points of intersection. APPLICATIONS INVOLVING STATIONARY POINTS If n epression cn be reduced to qudrtic function, the discriminnt my be used to determine the number of sttionry points on curve. When Δ > the function hs sttionry points. When Δ the function hs sttionry point. When Δ < the function hs no sttionry points. If sked to find the vlues of so tht, or no sttionry points eist nd eqution is qudrtic or disguised qudrtic  use the discriminnt. To show tht no sttionry points eist nd the derivtive is qudrtic epression show tht Δ <. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
3 MATRICES A mtri is rectngulr collection of numbers enclosed by round or squre brckets. The horizontl lines in mtri re referred to s rows nd the verticl lines re known s columns. A mtri with m rows nd n columns is clled m n mtri. Alterntively, we sy tht the order of mtri with m rows nd n columns is m n. i.e. The first number in the order indictes the number of rows. For emple: The mtri [ 9] is mtri. Two mtrices re equl if they re of the sme order nd if ll their corresponding elements re equl. For emple: If 9 9 then. MATRIX ARITHMETIC Like norml numbers, mtrices my be dded, subtrcted, multiplied nd divided. ADDITION AND SUBTRACTION OF MATRICES Mtrices cn only be dded nd/or subtrcted if they re of the sme dimensions. Addition of mtrices is performed by dding corresponding elements. c b e + d g f + e h c + g b + f d + h Subtrction of mtrices is performed by subtrcting corresponding elements. b e f e b f c d g h cg d h Mtri ddition is commuttive: A + B B + A Mtri ddition is ssocitive: A + ( B + C) ( A + B) + C In ddition: A+ O O+ A A A+ ( A) ( A) + A O Note tht O represents zero mtri ( mtri where every element is zero). Mtri subtrction is not commuttive: A B B A Mtri subtrction is not ssocitive: A ( BC) ( AB) C The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
4 The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge MULTIPLICATION BY A SCALAR The term sclr is used to represent rel number. When mtri is multiplied by sclr, we multiply ech element in the mtri by tht sclr. dk ck bk k d c b k Note: The mtri order is unchnged when mtri is multiplied by sclr. Sclr multipliction is distributive: ) ( B A k kb ka + + Sclr multipliction is ssocitive: A c k ca ka ) ( + + Note further tht: ) ( ) ( ca k A kc For emple: If A, B nd C then ) ( C B A
5 MATRIX MULTIPLICATION Mtri multipliction is possible only if the number of columns in the first mtri is the sme s the number of rows in the second mtri. The size of the resultnt mtri (the product) is equl to the number of rows in the first mtri nd the number of columns in the second mtri. ( m n)( n p) ( m p) When multiplying mtrices, we multiply the rows of the first mtri by the columns of the second mtri. For Emple: For Emple: For emple: Given tht A nd B ( ) + () + () AB ( ) + () + () () + ( ) + () () + ( ) + () Note: A is ( ) mtri, B is ( ) mtri, nd therefore, the product is ( ) mtri. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
6 In generl: Mtri multipliction is not commuttive: AB BA. (One eception is squre mtrices). Mtri multipliction is ssocitive: ( AB ) C A( BC) ( A + B) C AC + BC C ( A + B) CA + CB A ( B + C) AB + AC AO OA O The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge 6
7 THE DETERMINANT & MULTIPLICATIVE INVERSE (INVERSE MATRIX) When solving lgebric epressions of the form 6, we divide both sides by to solve for. The sme principles cn be pplied when solving mtri equtions tht require mtri tht is prt of product to be cncelled. For Emple: X 9 7 We use the multiplictive inverse or inverse under multipliction under these circumstnces. The multiplictive inverse of A is denoted s A nd hs the property tht: AA A A Given c b d the inverse mtri of is defined s: A d bc d c b where d bc is the determinnt of A For emple: Given Note: 7 B, det B () 7() The determinnt my be written s det A, A or b c d. The Determinnt of A is sclr quntity ( rel number). If the determinnt is zero ( d bc ) the inverse of the mtri does not eist. Such mtri is sid to be singulr nd hs no inverse. For emple: t t B t is singulr when t t i.e. When t,. A mtri is sid to be regulr if its inverse eists. Only squre mtrices cn hve inverses. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge 7
8 FINDING INVERSE MATRICES To Find n Inverse Mtri of the order without using technology: Given c b d the inverse mtri A where d d bc c d bc is the determinnt of A b METHOD: Step : Interchnge the elements on the leding digonl. Step : Chnge the sign of the other elements in the mtri. Step : Premultiply the mtri with For emple: d bc. Given C, find the inverse mtri C. Inverse will eist if det A : det A () (), therefore, A does eist. C Use technology to find the inverse of higher order mtrices. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge 8
9 SOLVING MATRIX EQUATIONS Just s lgebric epressions cn be simplified nd solved in the rel number system, mtri epressions cn lso be solved in the sme mnner. When solving mtri equtions, it is importnt to remember the following differences between mtri lgebr nd rel lgebr: The commuttive lw for multipliction is not obeyed by mtrices ( AB BA). It is not lwys possible to find the multiplictive inverse of mtri. Emples of common equtions to be solved: Let A denote the known mtri. Let X denote n unknown mtri (the mtri we re solving for). () Given AX B : Premultiply both sides by A ( A A I which isoltes X ). A AX A IX A B X A B B For emple: Find mtri X such tht X. X X 6 (b) Given XA B : Postmultiply both sides by A XAA BA XI BA s X BA For emple: AA I Find mtri X such tht X 6... X X The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge 9
10 ALGEBRA OF INDICIAL & EXPONENTIAL FUNCTIONS An indicil or eponentil epression is n lgebric epression involving number tht is rised to power. For Emple: is n indicil epression. The generl form for indicil equtions is written s n, where: represents the inde or power or eponent. represents the bse. n represents the bsic numerl. INDEX LAWS Indicil epressions my be simplified by pplying the following Inde Lws. MULTIPLYING INDICIAL EXPRESSIONS To multiply indicil epressions with the sme bse, dd the powers. m n m+ n For Emple: DIVIDING INDICIAL EXPRESSIONS To divide indicil epressions with the sme bse, subtrct the powers. m n mn For Emple: REMOVING BRACKETS To remove brckets round epressions, multiply the power on ech number nd pronumerl to the power on the brckets. m n mn ( ) 6 9 For Emple: ( ) y y 8 y The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
11 ZERO POWERS Any term or number (ecept for ), tht hs been rised to the power of zero is equl to. 6 8 For Emple: ( y z ) NEGATIVE POWERS When term is moved from the numertor to the denomintor or vice vers, the sign of the power chnges. n n, Indicil epressions with negtive powers my therefore be written with positive powers by moving terms from the numertor to the denomintor, nd vis vers. For Emple: 7 For Emple: b 7 7 b 9b In generl: RATIONAL EXPONENTS q Therefore, q p q q nd ( ) nd p q p For emple: ( ) The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
12 SIMPLIFYING INDICIAL EXPRESSIONS METHOD: Step : Remove brckets. Step : Apply the pproprite Inde Lws. Step : Rewrite the nswer with positive powers. To simplify indicil epressions, it my be necessry to pply more thn one inde lw. WATCHOUTS You cn only use inde lws when multiplying or dividing indices with the sme bse. Terms seprted by ddition or subtrction need to be simplified by pplying inde lws in reverse. For emple: As n m n m+ n then n m+ n m n Terms with different bses cnnot be simplified by pplying inde lws without pplying the chnge of bse rule. y For emple: No Solution For emple: ( ) y y y + y e Anything. Therefore, if you see the number  think Anything log e e ( 6 ) NOT The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
13 SOLVING INDICIAL EQUATIONS Recll tht the process of solving equtions in the form: epression loctes the X intercepts on the corresponding grph. epression epression loctes the points of intersection of the grphs yepression nd y epression. There re three different pproches tht my be pplied to solve indicil epressions: Equting powers. Rising epressions to reciprocl powers. Applying logrithms. Which method is to be pplied depends upon the mnner in which the eqution is presented, nd whether the unknown is the bsic numerl, bse or power. METHOD: Step : Apply the pproprite inde lws to reduce the given eqution to s few terms s possible. Step : Solve the eqution writing your nswer with positive powers. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
14 SOLUTION STRATEGIES IF THE EQUATION CAN BE REDUCED TO TERMS Write ech term on either side of the equlity sign nd solve. You will typiclly obtin one of the following eqution formts: () Indicil equtions where the power is unknown. Mnipulte the given eqution so tht the bses re the sme, nd then equte m n powers. i.e. If then m n. For emple: Solve n 8 n n n (b) Indicil equtions where the bse is unknown. If the unknown is the bse, rise both sides of the indicil eqution to the reciprocl of the power of the eisting inde. i.e. b b For emple: Solve for given tht. (). Note: 6 If the power on the bse is even (eg. ) OR if the numertor in the power is even (eg. ), the solutions for will be positive or negtive. For emple: ( ) ( ) ( ) ± ± (c) Indicil equtions tht cnnot be simplified. Tke log or log e of both sides of the eqution or solve using technology (this will be necessity for questions involving combined functions). For emple:. log log. log log. log. log.6 The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
15 OTHER SOLUTION STRATEGIES () Apply the Null Fctor Lw: Bring ll terms to one side of the eqution, fctorise nd pply the Null Fctor Lw. 6 For emple: e e e ( e ) e or e e e (Note: There is no solution for e ). (b) Assess whether the eqution is disguised qudrtic epression. Is one power twice the vlue of nother power? If yes, the epression could be disguised qudrtic. For emple: Solve ( ) + Let m m m+ ( m)( m ) m or m or or or For emple: Solve ( ) + ( ) + A + A+ where A Step : Cn I mke the bses the sme? SUMMARY OF SOLUTION PROCESSES If yes, use the following rule: If m. m n then n Step : Is the unknown the bse? If yes, use b b. Step : Cn I tke log or log e of both sides of the eqution? The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge
16 WATCHOUTS There re no vlues of the power for which n eponentil function without verticl trnsltion is less thn or equl to zero i.e. nd ve. For emple: cn t be solved. Answer is undefined. ve number y   If terms re seprted by multipliction or division, check to see if the bses re the sme. If yes, simplify epressions using Inde Lws. Powers consisting of two or more terms my be simplified by writing the eqution s series of terms which contin just one term in the power. Use the Inde Lws in reverse m+ n m n to chieve this i.e.. For emple: n+ n If terms re seprted by ddition nd subtrction YOU CANNOT pply Inde Lws directly. Use Inde Lws to write ech term s products nd remove common fctors n+ n n n n n For emple: + (.). Given n eponentil eqution with two eponentil terms nd one term is double the power of the other the eqution my be disguised qudrtic. Use Let A method. t t t e t e t t e. e + e + e + ( ) ( ) t + t e e, A + A etc t t e e Alwys check nswers by substitution or by finding the points of intersection using your clcultor. Alwys test the vlidity of your solutions by substituting the vlues of into the given eqution. If sked to solve n inequlity  proceed using n equlity sign. Once solutions hve been obtined, sketch grph nd use logic to determine the pproprite nswer. Test point before leving the question. When n inequlity is multiplied or divided by negtive number, the direction of the inequlity must be chnged. The School For Ecellence 6 The Essentils Unit Mthemticl Methods Reference Mterils Pge 6
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