Chapter-4-QUADRATIC EQUATION-(E)-01-THEORY

4.1 Polynomial. Algebraic expression containing many terms of the form , n being a non-negative integer is called a polynomial. i.e., , where x is a variable, are constants and Example : , . (1) Real polynomial : Let be real numbers and x is a real variable. Then is called real polynomial of real variable x with real coefficients. Example : etc. are real polynomials. (2) Complex polynomial : If be complex numbers and x is a varying complex number. Then is called complex polynomial of complex variable x with complex coefficients. Example : etc. are complex polynomials. (3) Degree of polynomial : Highest power of variable x in a polynomial is called degree of polynomial. Example : is a n degree polynomial. is a 3 degree polynomial. is single degree polynomial or linear polynomial. is an odd linear polynomial. A polynomial of second degree is generally called a quadratic polynomial. Polynomials of degree 3 and 4 are known as cubic and biquadratic polynomials respectively. (4) Polynomial equation : If f(x) is a polynomial, real or complex, then f(x) = 0 is called a polynomial equation. 4.2 Types of Quadratic Equation. A quadratic polynomial f(x) when equated to zero is called quadratic equation. Example : or An equation in which the highest power of the unknown quantity is two is called quadratic equation. Quadratic equations are of two types : (1) Purely quadratic equation : A quadratic equation in which the term containing the first degree of the unknown quantity is absent is called a purely quadratic equation. i.e. where a, c  C and a  0 (2) Adfected quadratic equation : A quadratic equation which contains terms of first as well as second degrees of the unknown quantity is called an adfected quadratic equation. i.e. where a, b, c  C and a  0, b  0. (3) Roots of a quadratic equation : The values of variable x which satisfy the quadratic equation is called roots of quadratic equation. Important Tips  An equation of degree n has n roots, real or imaginary.  Surd and imaginary roots always occur in pairs in a polynomial equation with real coefficients i.e. if 2 – 3i is a root of an equation, then 2 + 3i is also its root. Similarly if is a root of given equation, then is also its root.  An odd degree equation has at least one real root whose sign is opposite to that of its last term (constant term), provided that the coefficient of highest degree term is positive.  Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive has at least two real roots, one positive and one negative. 4.3 Solution of Quadratic Equation. (1) Factorization method : Let . Then and will satisfy the given equation. Hence, factorize the equation and equating each factor to zero gives roots of the equation. Example :  (2) Hindu method (Sri Dharacharya method) : By completing the perfect square as  Adding and subtracting , which gives, Hence the quadratic equation (a  0) has two roots, given by , Note :  Every quadratic equation has two and only two roots. 4.4 Nature of Roots. In quadratic equation , the term is called discriminant of the equation, which plays an important role in finding the nature of the roots. It is denoted by  or D. (1) If a, b, c  R and a  0, then : (i) If D < 0, then equation has non-real complex roots. (ii) If D > 0, then equation has real and distinct roots, namely , and then …..(i) (iii) If D = 0, then equation has real and equal roots and then …..(ii) To represent the quadratic expression in form (i) and (ii), transform it into linear factors. (iv) If D  0, then equation has real roots. (2) If a, b, c  Q, a  0, then : (i) If D > 0 and D is a perfect square  roots are unequal and rational. (ii) If D > 0 and D is not a perfect square  roots are irrational and unequal. (3) Conjugate roots : The irrational and complex roots of a quadratic equation always occur in pairs. Therefore (i) If one root be then other root will be . (ii) If one root be then other root will be . (4) If D1 and D2 be the discriminants of two quadratic equations,then (i) If , then (a) At least one of and . (b) If then (ii) If , then (a) At least one of and . (b) If then . 4.5 Roots Under Particular Conditions. For the quadratic equation . (1) If  roots are of equal magnitude but of opposite sign. (2) If  one root is zero, other is . (3) If  both roots are zero. (4) If  roots are reciprocal to each other. (5) If  roots are of opposite signs. (6) If  both roots are negative, provided . (7) If  both roots are positive, provided . (8) If sign of a = sign of b  sign of c  greater root in magnitude, is negative. (9) If sign of b = sign of c  sign of a  greater root in magnitude, is positive. (10) If  one root is 1 and second root is c/a. (11) If , then equation will become an identity and will be satisfied by every value of x. (12) If and b, c  I and the root of equation are rational numbers, then these roots must be integers. Important Tips  If an equation has only one change of sign, it has one +ve root and no more.  If all the terms of an equation are +ve and the equation involves no odd power of x, then all its roots are complex. Example: 1 Both the roots of given equation are always (a) Positive (b) Negative (c) Real (d) Imaginary Solution: (c) Given equation can be re-written as Hence both roots are always real. Example: 2 If the roots of are equal then (a) 2b (b) (c) 3b (d) b Solution: (a) Hence one root is 1. Also as roots are equal, other root will also be equal to 1. Also    Example: 3 If the roots of equation are equal in magnitude but opposite in sign, then (a) 2r (b) r (c) – 2r (d) None of these Solution: (a) Given equation can be written as Since the roots are equal and of opposite sign,  Sum of roots = 0   Example: 4 If 3 is a root of , it is also a root of (a) (b) (c) (d) Solution: (c) Equation has one root as 3,   Put and in option Only (c) gives the correct answer i.e.   Example: 5 For what values of k will the equation have equal roots (a) 1, –10/9 (b) 2, –10/9 (c) 3, –10/9 (d) 4, –10/9 Solution: (b) Since roots are equal then   Solving, we get 4.6 Relations between Roots and Coefficients. (1) Relation between roots and coefficients of quadratic equation : If  and  are the roots of quadratic equation , (a  0) then Sum of roots Product of roots If roots of quadratic equation (a  0) are  and  then (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (2) Formation of an equation with given roots : A quadratic equation whose roots are  and  is given by  i.e.  (3) Equation in terms of the roots of another equation : If ,  are roots of the equation , then the equation whose roots are (i) –, –  (Replace x by – x) (ii)  (Replace x by 1/x) (iii) ; n  N  (Replace x by ) (iv) k, k  (Replace x by x/k) (v) ,  (Replace x by (x – k)) (vi)  (Replace x by kx) (vii) ; n  N  (Replace x by ) (4) Symmetric expressions : The symmetric expressions of the roots ,  of an equation are those expressions in  and , which do not change by interchanging  and . To find the value of such an expression, we generally express that in terms of and . Some examples of symmetric expressions are : (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) 4.7 Biquadratic Equation. If , , ,  are roots of the biquadratic equation , then , or , or and Example: 6 If the difference between the corresponding roots of and is same and , then (a) (b) (c) (d) Solution: (a) ,  and ,  According to question,   Example: 7 If the sum of the roots of the quadratic equation is equal to the sum of the squares of their reciprocals, then are in (a) A.P. (b) G.P. (c) H.P. (d) None of these Solution: (c) As given, if ,  be the roots of the quadratic equation, then      are in A.P.  are in H.P. Example: 8 Let ,  be the roots of and ,  be root of . If , , ,  are in G.P., then the integral value of p and q respectively are (a) – 2, – 32 (b) – 2, 3 (c) – 6, 3 (d) – 6, – 32 Solution: (a) , , , Since , , ,  are in G.P.  ,  So  If ,  and ,  But   , Example: 9 If 1 – i is a root of the equation , then the values of a and b are (a) 2, 1 (b) – 2, 2 (c) 2, 2 (d) 2, – 2 Solution: (b) Since is a root of .  is also a root. Sum of roots   Product of roots   Hence , Example: 10 If the roots of the equation are ,  and the roots of equation are , , then (a) (b) (c) (d) Solution: (b) Since roots of the equation are .  and    and    Example: 11 If , but , then the equation whose roots are and is (a) (b) (c) (d) None of these Solution: (b) ,  . ,  are roots of . Therefore , ∵  Example: 12 Let ,  be the roots of the equation , , then the roots of the equation are (a) a, c (b) b, c (c) a, b (d) a, d Solution: (c) Since ,  are the roots of i.e. of   and   a, b are the roots of  Hence (c) is the correct answer Example: 13 If  and  are roots of the equation and , then (a) (b) (c) (d) Solution: (a) Since  and  are roots of equation, , therefore , Now,  Example: 14 If one root of the equation is the square of the other, then (a) (b) (c) (d) Solution: (d) Let  and be the roots then , Now   Example: 15 Let  and  be the roots of the equation , the equation whose roots are is (a) (b) (c) (d) Solution: (d) Roots of are Take   Required equation is Example: 16 If one root of a quadratic equation is , then the equation is (a) (b) (c) (d) None of these Solution: (b) Given root ,  other root Again, sum of roots = – 4 and product of roots = – 1. The required equation is 4.8 Condition for Common Roots. (1) Only one root is common : Let  be the common root of quadratic equations and .  , By Crammer’s rule : or  ,  The condition for only one root common is (2) Both roots are common: Then required condition is . Important Tips  To find the common root of two equations, make the coefficient of second degree term in the two equations equal and subtract. The value of x obtained is the required common root.  Two different quadratic equations with rational coefficient can not have single common root which is complex or irrational as imaginary and surd roots always occur in pair. Example: 17 If one of the roots of the equation and is coincident. Then the numerical value of is (a) 0 (b) – 1 (c) (d) 5 Solution: (b) If  is the coincident root, then and  ,   Example: 18 If a, b, c are in G.P. then the equations and have a common root if are in (a) A.P. (b) G.P. (c) H.P. (d) None of these Solution: (a) As given,  can be written as   This must be common root by hypothesis So it must satisfy the equation,   Hence are in A.P. 4.9 Properties of Quadratic Equation. (1) If f(a) and f(b) are of opposite signs then at least one or in general odd number of roots of the equation lie between a and b. (2) If then there exists a point c between a and b such that , . As is clear from the figure, in either case there is a point P or Q at where tangent is parallel to x-axis i.e. at . (3) If  is a root of the equation then the polynomial is exactly divisible by or is factor of . (4) If the roots of the quadratic equations , are in the same ratio then . (5) If one root is k times the other root of the quadratic equation then . Example: 19 The value of ‘a’ for which one root of the quadratic equation is twice as large as the other is (a) 2/3 (b) – 2/3 (c) 1/3 (d) – 1/3 Solution: (a) Let the roots are  and 2 Now, ,  ,      4.10 Quadratic Expression. An expression of the form , where a, b, c  R and a  0 is called a quadratic expression in x. So in general, quadratic expression is represented by or . (1) Graph of a quadratic expression : We have  Now, let and  (i) The graph of the curve is parabolic. (ii) The axis of parabola is or i.e. (parallel to y-axis). (iii) (a) If a > 0, then the parabola opens upward. (b) If a < 0, then the parabola opens downward. (iv) Intersection with axis (a) x-axis: For x axis,   For D > 0, parabola cuts x-axis in two real and distinct points i.e. . For D = 0, parabola touches x-axis in one point, . For D < 0, parabola does not cut x-axis(i.e. imaginary value of x). (b) y-axis : For y axis , (2) Maximum and minimum values of quadratic expression : Maximum and minimum value of quadratic expression can be found out by two methods : (i) Discriminant method : In a quadratic expression . (a) If a > 0, quadratic expression has least value at . This least value is given by . (b) If a < 0, quadratic expression has greatest value at . This greatest value is given by . (ii) Graphical method : Vertex of the parabola is , i.e., ,  , Hence, vertex of is (a) For a > 0, f(x) has least value at . This least value is given by . (b) For a < 0, f(x) has greatest value at . This greatest value is given by . (3) Sign of quadratic expression : Let or Where a, b, c  R and a  0, for some values of x, f(x) may be positive, negative or zero. This gives the following cases : (i) a > 0 and D < 0, so for all i.e., is positive for all real values of x. (ii) a < 0 and D < 0, so for all x  R i.e., f(x) is negative for all real values of x. (iii) a > 0 and D = 0 so, for all x  R i.e., f(x) is positive for all real values of x except at vertex, where . (iv) a < 0 and D = 0 so, for all x  R i.e. f(x) is negative for all real values of x except at vertex, where . (v) a > 0 and D > 0 Let have two real roots  and  , then for all and for all . (vi) and Let have two real roots  and  , Then for all and for all Example: 20 If x be real, then the minimum value of is (a) – 1 (b) 0 (c) 1 (d) 2 Solution: (c) Since therefore its minimum value is Example: 21 If x is real, then greatest and least values of are (a) 3, –1/2 (b) 3, 1/3 (c) – 3, –1/3 (d) None of these Solution: (b) Let ∵ x is real, therefore      Thus greatest and least values of expression are 3, 1/3 respectively. Example: 22 If f(x) is quadratic expression which is positive for all real value of x and . Then for any real value of x (a) (b) (c) (d) Solution: (b) Let , then ∵ . Therefore and Now for g(x), Discriminant as Therefore sign of g(x) and a are same i.e. . Example: 23 If ,  are roots of the equation where then (a) (b) (c) (d) Solution: (b) Since  Roots will be of opposite sign, (b > 0) It is given that So, is possible only when   4.11 Wavy Curve Method. Let …..(i) Where and are fixed natural numbers satisfying the condition First we mark the numbers on the real axis and the plus sign in the interval of the right of the largest of these numbers, i.e. on the right of . If is even then we put plus sign on the left of and if is odd then we put minus sign on the left of . In the next interval we put a sign according to the following rule : When passing through the point the polynomial f(x) changes sign if is an odd number and the polynomial f(x) has same sign if is an even number. Then, we consider the next interval and put a sign in it using the same rule. Thus, we consider all the intervals. The solution of is the union of all intervals in which we have put the plus sign and the solution of is the union of all intervals in which we have put the minus sign. 4.12 Position of Roots of a Quadratic Equation. Let , where a, b, c  R be a quadratic expression and be real numbers such that . Let ,  be the roots of the equation i.e. . Then , where D is the discriminant of the equation. (1) Condition for a number k (If both the roots of f(x) = 0 are less than k) (i) (roots may be equal) (ii) (iii) , where (2) Condition for a number k (If both the roots of f(x) = 0 are greater than k) (i) (roots may be equal) (ii) (iii) , where (3) Condition for a number k (If k lies between the roots of f(x) = 0) (i) (ii) , where (4) Condition for numbers k1 and k2 (If exactly one root of f(x) = 0 lies in the interval (k1, k2)) (i) (ii) , where . (5) Condition for numbers k1 and k2 (If both roots of f(x) = 0 are confined between k1 and k2) (i) (roots may be equal) (ii) (iii) (iv) , where and (6) Condition for numbers k1 and k2 (If k1 and k2 lie between the roots of f(x) = 0) (i) (ii) (iii) , where Example: 24 If the roots of the equation are real and less than 3, then (a) a < 2 (b) (c) (d) Solution: (a) Given equation is If roots are real, then     As roots are less than 3, hence    . Hence a < 2 satisfy all the conditions. Example: 25 The value of a for which may have one root less than a and another root greater than a are given by (a) (b) (c) (d) or Solution: (d) The given condition suggest that a lies between the roots. Let For ‘a’ to lie between the roots we must have Discriminant  0 and Now, Discriminant  0  which is always true. Also      or 4.13 Descarte's Rule of Signs. The maximum number of positive real roots of a polynomial equation is the number of changes of sign from positive to negative and negative to positive in f(x). The maximum number of negative real roots of a polynomial equation is the number of changes of sign from positive to negative and negative to positive in . Example: 26 The maximum possible number of real roots of equation is (a) 0 (b) 3 (c) 4 (d) 5 Solution: (b) + – – + 2 changes of sign  maximum two positive roots. – – + + 1 changes of sign  maximum one negative roots.  total maximum possible number of real roots = 2 + 1 = 3. 4.14 Rational Algebraic Inequations. (1) Values of rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions : To find the values attained by rational expression of the form for real values of x, the following algorithm will explain the procedure : Algorithm Step I: Equate the given rational expression to y. Step II: Obtain a quadratic equation in x by simplifying the expression in step I. Step III: Obtain the discriminant of the quadratic equation in Step II. Step IV: Put Discriminant  0 and solve the inequation for y. The values of y so obtained determines the set of values attained by the given rational expression. (2) Solution of rational algebraic inequation: If P(x) and Q(x) are polynomial in x, then the inequation and are known as rational algebraic inequations. To solve these inequations we use the sign method as explained in the following algorithm. Algorithm Step I: Obtain P(x) and Q(x). Step II: Factorize P(x) and Q(x) into linear factors. Step III: Make the coefficient of x positive in all factors. Step IV: Obtain critical points by equating all factors to zero. Step V: Plot the critical points on the number line. If there are n critical points, they divide the number line into (n + 1) regions. Step VI: In the right most region the expression bears positive sign and in other regions the expression bears positive and negative signs depending on the exponents of the factors. 4.15 Algebraic Interpretation of Rolle’s Theorem. Let f(x) be a polynomial having  and  as its roots, such that . Then, . Also a polynomial function is everywhere continuous and differentiable. Thus f(x) satisfies all the three conditions of Rolle’s theorem. Consequently there exists such that i.e. at . In other words is a root of . Thus algebraically Rolle’s theorem can be interpreted as follows. Between any two roots of polynomial f(x), there is always a root of its derivative . Lagrange’s theorem : Let f(x) be a function defined on [a b] such that (i) f(x) is continuous on [a b] and (ii) f(x) is differentiable on (a, b), then c  (a, b), such that Lagrange’s identity : If then : Example: 27 If , then (a) (b) (c) (d) Solution: (c) Given     Equating each factor equal to 0, We get   or Example: 28 If for real values of x, and , then (a) (b) (c) or (d) Solution: (c) or  …..(i) Again or  …..(ii) From eq. (i) and (ii),  or Example: 29 If , then the equation , has (a) Both roots in [a b] (b) Both roots in (– , a) (c) Both roots in (b, ) (d) One root in (– , a) and other in (b, +) Solution: (d) We have,  [∵ b > a] , i.e. and (b, ). Example: 30 The number of integral solution of is (a) 1 (b) 2 (c) 5 (d) None of these Solution: (c)    Approximately, Hence, integral values of x are 0, 1, 2, 3, 4 Hence, number of integral solution = 5 Example: 31 If then at least one root of the equation lies in the interval (a) (0, 1) (b) (1, 2) (c) (2, 3) (d) (3, 4) Solution: (a) Let  Clearly , Since, . Hence, there exists at least one point c in between 0 and 1, such that , by Rolle’s theorem. Trick: Put the value of in given equation   , which lie in the interval (0, 1) 4.16 Equation and Inequation containing Absolute Value. (1) Equations containing absolute values By definition, Important forms containing absolute value : Form I: The equation of the form is equivalent of the system . Form II: The equation of the form …..(i) Where are functions of x and g(x) may be a constant. Equations of this form can be solved by the method of interval. We first find all critical points of . If coefficient of x is +ve, then graph starts with +ve sign and if it is negative, then graph starts with negative sign. Then using the definition of the absolute value, we pass form equation (i) to a collection of system which do not contain the absolute value symbols. (2) Inequations containing absolute value By definition, |x| < a  (a > 0),  , |x| > a  or and  or Example: 32 The roots of are (a) 0, 4 (b) –1, 3 (c) 4, 2 (d) 5, 1 Solution: (a) We have Let   and  and , which is not possible.  or  or Example: 33 The set of all real numbers x for which , is (a) (b) (c) (d) Solution: (b) Case I: If i.e. , we get    But  …..(i) Case II: i.e. , then   . Which is true for all x  …..(ii) From (i) and (ii), we get, Example: 34 Product of real roots of the equation (t  0) (a) Is always +ve (b) Is always –ve (c) Does not exist (d) None of these Solution: (c) Expression is always +ve, so . Hence roots of given equation does not exist. Example: 35 The number of solution of (a) 3 (b) 1 (c) 2 (d) 0 Solution: (b) We have    or But , when .  Only solution is . Hence number of solution is one.