QUADRATIC EQUATION & EXPRESSIONS

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1 QUADRATIC EQUATION & EXPRESSIONS Qudrtic Eqution An eqution of the form x + bx + c 0, where 0 nd, b, c re rel numbers, is clled qudrtic eqution. The numbers, b, c re clled the coefficients of the qudrtic eqution. A root of the qudrtic eqution is number α (rel or complex) such tht α + bα + c 0. The roots of the qudrtic eqution re given by x Bsic Results: b± b 4c The quntity D (Db 4c) is known s the discriminnt of the qudrtic eqution. The qudrtic eqution hs rel nd equl roots if nd only if D 0 i.e. b 4c 0. The qudrtic eqution hs rel nd distinct roots if nd only if D > 0 i.e. b 4c > 0. The qudrtic eqution hs complex roots with non zero imginry prts if nd only if D < 0 i.e. b 4c < 0. Let α nd β be two roots of the given qudrtic eqution. Then α + β b/ nd αβ c/. If α nd β re the roots of n eqution, then the eqution is x (α + β) x + αβ 0. In qudrtic eqution x + bx + c 0 if + b + c 0, then x 1 is one of the root of the eqution. If the qudrtic eqution x + bx + c 0 is stisfied by more thn two numbers (rel or imginry) then it becomes n identity i.e., b c 0 x + 1 x Illustrtion 1: Solve + (x 1, - ). x 1 x +. x+ 1 x + x 1 x+ x + 1 x + 0 x 1 x+ x+ 1 x+ + x 1 x x 1 x+ x 1 x+ ( )( ) ( )( ) ( )( ) ( )( ) 0 (x + 1) (x + ) + (x 1) (x ) (x 1) (x + ) 0 x + x + + x x + (x + x ) 0 x x (x + x 10) 0 x + x 10 0 (x + 5) (x ) 0 x 5, or x. x 5, re the solutions of the given eqution. Illustrtion : Find if the roots of the eqution ( + b c) x + ( + c b) x + (b + c ) 0 re rel or complex, where, b, c R. Since ( + b c) + ( + c b) + (b + c ) 0 x 1 is one of the root. FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

2 PDT Coursewre 10 th Moving to 11 th MA-QEE- Hence the roots re rel. Illustrtion :. The sum of squres of two consecutive positive integers is 1. Find the integers. Let x be one of the positive integers. Then the other is x + 1 sum of squres of the integers x + (x + 1) 1 x + x + x x + x 0 0 (x + x 110) 0 x + x (x 10) (x + 11) 0 x 10 or x 11. consecutive positive integers re 10 nd 11. Illustrtion 4:. One side of rectngle exceeds its other side by cm. If its re is 195 cm, determine the sides of the rectngle. Let one side be x cm Then other side will be (x + ) cm Are of rectngle x (x + ) x (x + ) 195 x + x x + 15x 1x x(x + 15) 1(x + 15) 0 (x 1) (x + 15) 0 x 1 or x 15. Since, side of the rectngle cnnot be negtive x 1. sides of rectngle re x, x + 1 cm, 15 cm. Illustrtion 5: The hypotenuse of right ngled tringle is 5 cm. The difference between the lengths of the other two sides of the tringle is 17 cm. Find the lengths of these sides. Let the length of the shorter side be x cm. Then, the length of the longer side (x + 17) cm AB x, BC x + 17, CA 5 By Pythgors theorem AB + BC 5 x + (x + 17) 5 x + 4x 6 0 x + 17x (x 7) (x + 4) 0 x 7, x 4 But side of tringle cnnot be negtive x 7 Length of the shorter side x 7 cm Let of the longer side x cm. x A B 5 x+17 C FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

3 MA-QEE- PDT Coursewre 10 th Moving to 11 th 4 5 Illustrtion 6: Solve the following eqution by fctoriztion method x x + x 0,. 4 5 x x+ (4 x) (x + ) 5x 6x + 6x 1 0 x + x 0 (x + ) (x 1) 0 x or x 1. Illustrtion 7: Find k for which the eqution (k 1)x + (k 1) x + 0 hs equl roots. For the given eqution k 1, b (k 1) c D b 4c 4(k 1) 4(k 1) () 4(k 1) (k 14) A qudrtic eqution hs equl roots if D 0 4(k 1) (k 14) 0 k 1 or k 14 k 14 only. Illustrtion 8: If the eqution ( ) ( ) possible vlues of m. x + m x + m 4m+ 4 0 hs coincident roots, then find the roots re coincident, discriminnt 0 ( m) 4( m 4m 4) + +. This will give m or 6. Illustrtion 9: Solve x + -x 0. x + x 0 x let t x x t + 1 t 0 t t (t 1) 0 t 1 x 1 x 0 x 0. Illustrtion 10: If the eqution (λ 5λ + 6 )x + (λ λ + )x + (λ 4) 0 is stisfied by more thn two vlues of x, then find the prmeter λ. FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

4 PDT Coursewre 10 th Moving to 11 th MA-QEE-4 If n eqution of degree two is stisfied by more thn two vlues of x, then it must be n identity. λ 5λ + 6 0, λ λ + 0, λ 4 0 λ, nd λ 1, nd λ,. Common vlue of λ which stisfies ech condition is λ. Illustrtion 11: Find the vlue of m for which the eqution (1 + m)x (1 + m)x + (1 + 8m) 0 hs equl roots. Given eqution is (1 + m)x (1 + m)x + (1 + 8m) 0. If roots re equl, then discriminnt 4(1 + m) 4(1 + m) (1 + 8m) 0 m m 0 or m 0,. Problems bsed on sum nd product of roots Illustrtion 1: Find the vlue of m if the product of the roots of the eqution mx + 6x + (m 1) 0 is 1. Product of the roots m 1 1 m m 1 m m 1 m 1. Illustrtion 1: If the sum of the roots of the eqution qx + x + q 0 is equl to their product, then find the vlue of q. Let α, β be the roots α + β q α β q q Since α + β α β q q Illustrtion 14: Determine the eqution, sum of whose roots is 1 nd sum of their squres is 1. Let α, β be roots of the required qudrtic eqution α + β 1, α + β 1 (α+ β) α + β + αβ αβ αβ -1 αβ -6. required qudrtic eqution is x (α + β)x + αβ 0 FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

5 MA-QEE-5 PDT Coursewre 10 th Moving to 11 th x x 6 0. Illustrtion 15: If the roots of the eqution x + bx + c 0 re in the rtio m : n, prove tht mnb c(m + n) α : β m : n α m β n α+β m+ n (Applying Componendo nd Dividendo) α β m n α+β ( m+ n) ( α β) ( m n) ( α+β ) ( m+ n) α+β 4αβ m n ( ) ( ) ( m+ n) ( ) b / b c 4 m n b m+ n b 4c m n ( ) ( ) ( ) b (m n) b (m + n) 4c( m + n) 4c( m + n) b [(m + n) (m n) ] 4c(m + n) b [4mn] mnb c(m + n) Illustrtion 16: If α nd β be roots of eqution x α β α β + bx + c 0, prove tht + + b + β α β α Here α + β b/ nd αβ c/. α β α β Now, + + b + β α β α ( α +β ) + b( α +β ) αβ ( ) ( ) ( ) α+β αβ α+β + b α+β αβ αβ b c b b c b + c/ bc b c/. b. FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

6 PDT Coursewre 10 th Moving to 11 th MA-QEE-6 Illustrtion 17: If α nd β re the roots of the eqution x x 6 0, then find the eqution whose roots re α +, β +. Here α + β /, αβ 6/ so tht S α + β + 4 (α + β) αβ , P α β + (α + β ) + 4 α β [(α + β) αβ ] Therefore, the eqution is x x x 49x Illustrtion 18: If the roots of the eqution x px + q 0 differ by unity then find the reltion between p nd q. Suppose the eqution x px + q 0 hs the roots α + 1 nd α then α + 1+ α p α p (1) nd (α+1) α q α + α q () Putting the vlue of α from (1) in (), we get ( p 1) p 1 + q 4 (p 1) + (p 1) 4q p 1 4q p 4q + 1. Alterntive: Let α nd β be the roots. α β 1 (α + β) 4αβ 1 p 4q 1, or p 1+ 4q. Illustrtion 19: If p nd q re the roots of the eqution x + px + q 0, then find the vlue of p nd q. Since p nd q re roots of the eqution x + px + q 0, p + q p nd pq q pq q q 0 or p 1. If q 0, then p 0 nd if p 1, then q. Illustrtion 0: (α, β); (β, γ) nd (γ, α) re respectively the roots of x px + 0, x qx + 0 nd x rx If α, β nd γ re ll positive, then find the vlue of p + q + r. We hve, αβ, βγ nd γα 6, then αβ βγ γα 6 αβγ 6 nd α, β, γ re ll positive. Thus we hve α, β 1, γ Also, p α + β, q β + γ 4 nd r γ + α 5. Hence p + q + r 6. FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

7 MA-QEE-7 PDT Coursewre 10 th Moving to 11 th ASSIGNMENTS 1. Show tht if the roots of the eqution ( + b )x + x (c + bd) + c + d 0 re rel, they will be equl.. Prove tht the roots of the qudrtic eqution x bx 4 0 re rel nd distinct for ll rel nd b, where b.. Prove tht the roots of the eqution bx + (b c)x + (b c ) 0 re rel if those of x + bx + b 0 re imginry. 4. If α, β re the roots of the eqution x px + q 0 nd α 1, β 1 be the roots of the eqution x qx p 0, then find + αβ 1 αβ αα1 ββ If b R +, then show tht eqution ( ) ( ) ( ) + b x + + b x + 4+ b 0 hs no rel roots. 6. Find the rel roots of x 7x Let x (m )x + m 0 (m R) be qudrtic eqution. Find the vlue of m for which the roots (i) re equl, (ii) re opposite in sign, 8. If the roots of the eqution (x ) (x b) k 0 re c nd d, then prove tht the roots of (x c) (x d) + k 0 re nd b. 9. The coefficient of x in the qudrtic eqution x + px + q 0 ws tken s 17 in plce of 1. Its roots were found to be, nd 15. Find the roots of the originl eqution. 10. If α, β, γ re three distinct roots of the eqution ( )x + ( + b)x + (c 1) 0, then find the vlue of ( )5 + ( + b) 5 + (c 1). 11. If the roots of the eqution (b c)x + (c )x + ( b) 0 be equl, then prove tht, b, c re in rithmetic progression. 1. Find the number of rel roots of the eqution (x 1) + (x ) + (x ) If sum of roots of the eqution x + ( + ) x + b 0,, b R is, then find the vlue of. 14. If one root of the eqution x x k 0 be squre of the other, then find the vlue of k. 15. If the sum of the roots of the eqution ( + 1)x + ( + )x + ( + 4) 0 is 1, find the product of the roots. 16. If sinα, cosα re the roots of eqution cx + bx + 0, then show tht b c c If α, β be the roots of the eqution x 6x + 4 0, then find the vlue of α β α β αβ. β α β α α β FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

8 PDT Coursewre 10 th Moving to 11 th MA-QEE Let, b, c be three distinct positive rel numbers, then find the number of rel roots of x 4 + bx + c If x is the root of the eqution x + bx + c 0, then find the vlue of + b + c. 0. Solve x 1 x x x+ x x+ ( )( ) 1. Find the solutions set of x 4 5x If α, β re the roots of the eqution x + px + q 0, then find out the qudrtic eqution whose α β roots re 1 +,1+ β α.. If x nd x re the roots of the eqution x + 7x + b 0, find the vlues of nd b. 4. If one root is the squre of the other root of eqution x + px + q 0, then find the reltion between p nd q. 5. If the qudrtic eqution x + bx + c 0 hs the roots 0 nd 1, then find the vlue of + b. FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194

9 MA-QEE-9 PDT Coursewre 10 th Moving to 11 th ANSWERS TO ASSIGNMENTS 4. p q x p q x + p + q 4pq ,1 7. (i) m {1, 9} (ii) m < 0 9., zero ( ± 5) b c c x 1 1. x { ±, ± }. qx p x + p 0., 6 4. p (p 1) q + q FIITJEE Ltd., FIITJEE House, 9-A, Klu Sri, Srvpriy Vihr, New Delhi , Ph , , Fx 65194