CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients of the equation be able to manipulate expressions involving and be able to form equations with roots related to a given quadratic equation. In this chapter you will be looking at quadratic equations with particular emphasis on the properties of their solutions or roots.. The relationships between the roots and coefficients of a quadratic equation As you have already seen in the C module, any quadratic equation will have two roots (even though one may be a repeated root or the roots may not even be real). In this section you will be considering some further properties of these two roots. Suppose you know that the two solutions of a quadratic equation are x and x 5 and you want to find a quadratic equation having and 5 as its roots. The method consists of working backwards, i.e. following the steps for solving a quadratic equation but in reverse order. Now if x and x 5 are the solutions then the equation could have been factorised as (x )(x 5) 0. Expanding the brackets gives x 3x 0 0. This is a quadratic equation with roots and 5. Actually, any multiple of this equation will also have the same roots, e.g. x 3 6x 0 0 3x 9x 30 0 x 3 x 5 0
Roots of quadratic equations The general case Consider the most general quadratic equation ax bx c 0 and suppose that the two solutions are x and x. Now if and are the roots of the equation then you can work backwards to generate the original equation. A quadratic with the two solutions x and x is (x )(x ) 0. Expanding the brackets gives x x x 0 x ( )x 0 [] The most general quadratic equation is ax bx c 0 and this can easily be written in the same form as equation []. You can divide ax bx c 0 throughout by a giving Any multiple of this equation such as kx k( )x k 0 will also have roots and. x b a x c 0 [] a The coefficients of x in [] and in [] are now both equal to. Since [] and [] have the same roots, and, and are in the same form, you can write x ( )x x b a x c a Equating coefficients of x gives ( ) b a b a Equating constant terms gives Recall that the coefficient of x is the number in front of x. The sign means identically equal to. The coefficients of x and the constant terms must be equal. a c When the quadratic equation ax bx c 0 has roots and : The sum of the roots, b a ; and the product of roots, a c. Notice it is fairly easy to express x b a x c 0 as a x (sum of roots)x (product of roots) 0
Roots of quadratic equations 3 Worked example. Write down the sum and the product of the roots for each of the following equations: (a) x x 3 0, (b) x 8x 5 0. (a) x x 3 0 Here a, b and c 3. The sum of roots, b a 6. c 3 The product of roots, a. (b) x 8x 5 0 Here a, b 8 and c 5. The sum of roots, b a 8 8. c 5 The product of roots, 5. a Worked example. Find the sum and the product of the roots of each of the following quadratic equations: (a) 4x 8x 5, (b) x(x 4) 6 x. Take careful note of the signs. Notice the double negative. Neither equation is in the form ax bx c 0 and so the first thing to do is to get them into this standard form. (a) 4x 8x 5 4x 8x 5 0 In this case a 4, b 8 and c 5. The sum of roots, b a 8. 4 c 5 The product of roots, 5 a 4 4. (b) x(x 4) 6 x Expand the brackets and take everything onto the LHS. x 4x x 6 0 x x 6 0 Here a, b and c 6. The sum of roots, b a. c 6 The product of roots, 6. a Now in the form ax bx c 0. Now in the standard form.
4 Roots of quadratic equations Worked example.3 Write down equations with integer coefficients for which: (a) sum of roots 4, product of roots 7, (b) sum of roots 4, product of roots 5, (c) sum of roots 3 5, product of roots. (a) A quadratic equation can be written as x (sum of roots)x (product of roots) 0 x (4)x (7) 0 x 4x 7 0 (b) Using x (sum of roots)x (product of roots) 0 gives x (4)x (5) 0 x 4x 5 0 (c) Using x (sum of roots)x (product of roots) 0 gives x ( 3 5 )x ( ) 0 x 3 5 x 0 0x 0( 3 5 )x 0( ) 0 0x 6x 5 0 Again great care must be taken with the signs. Some of the coefficients are fractions not integers. You can eliminate the fractions by multiplying throughout by 0 or ( 5). You now have integer coefficients. EXERCISE A Find the sum and product of the roots for each of the following quadratic equations: (a) x 4x 9 0 (b) x 3x 5 0 (c) x 0x 3 0 (d) x 3x 0 (e) 7x x 6 (f) x(x ) x 6 (g) x(3 x) 5x (h) ax a x a 3 0 4 (i) ax 8a ( a)x (j) x 3 x 5 Write down a quadratic equation with: (a) sum of roots 5, product of roots 8, (b) sum of roots 3, product of roots 5, (c) sum of roots 4, product of roots 7, (d) sum of roots 9, product of roots 4, (e) sum of roots 4, product of roots 5, (f) sum of roots 3, product of roots 4, (g) sum of roots 3 5, product of roots 0, (h) sum of roots k, product of roots 3k, (i) sum of roots k, product of roots 6 k, (j) sum of roots ( a ), product of roots (a 7).
Roots of quadratic equations 5. Manipulating expressions involving and As you have seen in the last section, given a quadratic equation with roots and you can find the values of and without solving the equation. As you will see in this section, it is useful to be able to write other expressions involving and in terms of and. Worked example.4 Given that 4 and 7, find the values of: (a), (b). (a) You need to write this expression in terms of and in order to use the values given in the question. Now 4 7 (b) () 7 49 Two relations which will prove very useful are ( ) 3 3 ( ) 3 3( ) Notice how the expressions on the RHS contain combinations of just and. These two results can be proved fairly easily: ( + ) = ( + )( + ) ( + ) = + + + = ( + ) as required and ( ) 3 ( )( )( ) ( ) 3 ( )( ) ( ) 3 3 3 ( ) 3 3 3 3 3 ( ) 3 3 3 3( ) 3 3 ( ) 3 3( ) as required Take 3 as a factor.
6 Roots of quadratic equations Worked example.5 Given that 5 and, find the values of: (a), (b) 3 3, (c). (a) + = + = ( + ) Substitute the known values of and 5 () 5 4 9 Now it is in terms of and. (b) 3 3 ( ) 3 3( ) 3 3 5 3 3()(5) 5 30 55 (c) ( ) ( ) 5 ( ( ) ) 5 4 4 9 4 Worked example.6 Given that 5 and 3, find the value of ( ). ( ) ( ) ( ) 4 ( ) 5 4( 3 ) 5 8 3 3 Worked example.7 Write the expression in terms of and. 3 3 ( )3 3( ) ()
Roots of quadratic equations 7 EXERCISE B Write each of the following expressions in terms of and : (a) (b) (c) 3 3 (d) (e) ( )( ) (f) 5 5 Given that 3 and 9, find the values of: (a) 3 3, (b). 3 Given that 4 and 0, find the values of: (a), (b) 3 + 3. 4 Given that 7 and, find the values of: (a), (b). 5 The roots of the quadratic equation x 5x 3 0 are and. (a) Write down the values of and. (b) Hence find the values of: (i) ( 3)( 3), (ii). 6 The roots of the equation x 4x 3 0 are and. Without solving the equation, find the value of: (a) (b) (c) (d) (e) ( ) (f) ( )( ) 7 The roots of the quadratic equation x 4x 0 are and. (a) Find the values of: (i), (ii). (b) Hence find the value of: (i) ( )( ), (ii)..3 Forming new equations with related roots It is often possible to find a quadratic equation whose roots are related in some way to the roots of another given quadratic equation.
8 Roots of quadratic equations Worked example.8 The roots of the equation x 5x 6 0 are and. Find a quadratic equation whose roots are and. x 5x 6 0 The sum of roots, b a 5 5. c 6 The product of roots, 3. a You would be able to write down the new equation with roots and if you could find the sum and the product of the new roots. The sum of new roots, 5 6. The product of new roots, 3 3. Using x (sum of roots)x (product of roots) 0 the equation with roots and is x 5 6 x 3 0 5 3 x 5 6 x 0 3 6x 5x 0. The last example illustrates the basic method for forming new equations with roots that are related to the roots of a given equation: Multiply through by 6 in order to obtain integer coefficients (because it is often required in an examination question). Write down the sum of the roots,, and the product of the roots,, of the given equation. Find the sum and product of the new roots in terms of and. 3 Write down the new equation using x (sum of new roots)x (product of new roots) 0.
Roots of quadratic equations 9 Worked example.9 The roots of the equation x 7x 0 are and. Find the values of and. Hence, find a quadratic equation whose roots are and. From the equation x 7x 0 you have The sum of roots, b a 7 7. c The product of roots,. a Now, ( ) (7) () 49 4 53 and () () 4. Now since the roots of the new equation are and, and are the sum and product of the new roots. The required equation is x (sum of new roots)x (product of new roots) 0 x 53x 4 0. You could use any variable you choose. For instance, you could write y 53y 4 0. Worked example.0 The roots of the quadratic equation x 5x 3 0 are and. Find a quadratic equation whose roots are 3 and 3. From the equation x 5x 3 0 you have The sum of roots, b a 5 5. c 3 The product of roots, 3. a The new equation has roots 3 and 3. The sum of new roots, 3 3 ( ) 3 3( ) (5) 3 3(3)(5) 5 45 70. The product of new roots, 3 3 () 3 (3) 3 7. Using x (sum of new roots)x (product of new roots) 0 the required equation is x 70x 7 0.
0 Roots of quadratic equations EXERCISE C The roots of the equation x 6x 4 0 are and. Find a quadratic equation whose roots are and. The roots of the equation x 3x 5 0 are and. Find a quadratic equation whose roots are and. 3 The roots of the equation x 9x 5 0 are and. Find a quadratic equation whose roots are and. 4 Given that the roots of the equation x 5x 3 0 are and, find an equation with integer coefficients whose roots are and. 5 Given that the roots of the equation 3x 6x 0 are and, find a quadratic equation with integer coefficients whose roots are 3 and 3. 6 The roots of the equation x 4x 0 are and. Find a quadratic equation with integer coefficients whose roots are and. 7 The roots of the equation 3x 6x 0 are and. (a) Find the value of. (b) Find a quadratic equation whose roots are and. 8 The roots of the equation x 7x 3 0 are and. Without solving this equation, (a) find the value of 3 3. (b) Hence, find a quadratic equation with integer coefficients which has roots and. 9 Given that and are the roots of the equation 5x x 4 0, find a quadratic equation with integer coefficients which has roots and. 0 The roots of the equation x 4x 6 0 are and. Find an equation whose roots are and..4 Further examples Sometimes the questions may require you to apply similar techniques to the previous section but without directing you to find the sum and product of the roots.
Roots of quadratic equations Worked example. The roots of the equation x kx 8 0 are and 3. Find the two possible values of k. From x kx 8 0 you have The sum of roots, ( 3) k 3 k [] The product of roots, ( + 3) = 8 + 3 = 8 [] Equations [] and [] form a pair of simultaneous equations. From [] you have k 3 and substituting this into [] gives k 3 3 k 3 8 k 6k 9 3k 9 8 4 k 6k 9 6k 8 k the two possible values for k are and..5 New equations by means of a substitution Worked example.8 is reproduced below. The roots of the equation x 5x 6 0 are and. Find a quadratic equation whose roots are and. Since the new equation has roots that are the reciprocals of the original equation, the new equation can be found very quickly by making the substitution y x x, which transforms y x 5x 6 0 into y 5 6 0. y Multiplying throughout by y gives 5y 6y 0 or 6y 5y 0. Check the working in Worked example.8 to see which is quicker.
Roots of quadratic equations Worked example. The roots of the equation x 3x 5 0 are and. Find quadratic equations with roots (a) 3 and 3, (b) and. (a) Use the substitution y x 3 in x 3x 5 0, since the roots in the new equation are 3 less than those in the original equation. y x 3 x y 3, so new equation is (y 3) 3(y 3) 5 0 or y 6y 9 3y 9 5 0 The new equation is y 3y 5 0. (b) This time you need to use the substitution y x in x 3x 5 0 to eliminate x. Hence, y 3x 5 0 or y 5 3x. Squaring both sides gives (y 5) 9x 9y. Hence, y 0y 5 9y or y 9y 5 0. You could solve this question using the sum and product of roots technique of the previous section if you prefer. If you prefer, you can write the new equation as x 3x 5 0 or in terms of any other variable. Alternative solution Using 3 and 5 (a) Sum of new roots is ( 3) ( 3) 6 3 Product of new roots is ( 3)( 3) 3 3 9 3( ) 9 5 Hence new equation is y 3y 5 0. (b) Sum of new roots is ( ) 9 0 9 Product of new roots is () 5 Hence new equation is y 9y 5 0. You should see that the answers are the same whichever method you use. If the question directs you to use a particular substitution, e.g. Use the substitution y x to find a new equation whose roots are and, you must use the first method. However, if the question is of the form Use the substitution y x, or otherwise, to find a new equation whose roots are and, or simply Find an equation whose roots are and you may use either technique.
Roots of quadratic equations 3 Worked examination question.3 The roots of the quadratic equation x 3x 7 0 are and. (a) Write down the values of (i), (ii). (b) Find a quadratic equation with integer coefficients whose roots are and. (a) From the equation x 3x 7 0 you have The sum of roots, b a 3 3. c 7 The product of roots, 7. a (b) The new equation has roots and. The sum of new roots, ( ) 3 (7) 7 9 4 7 3 7 The product of new roots,. Using x (sum of new roots)x (product of new roots) 0 the required equation is x 3 x 0 7 x 3 x 0 7 7x 3x 7 0 A common mistake is to forget to multiply throughout by 7 to obtain integer coefficients. EXERCISE D The roots of the equation x 9x k 0 are and. Find the value of k. The roots of the quadratic equation 4x kx 5 0 are and 3. Find the two possible values of the constant k. 3 The roots of the quadratic equation x 3x 7 0 are and. Find an equation with integer coefficients which has roots 3 3 and. 4 The roots of the quadratic equation x 5x 0 are and. Find an equation with integer coefficients which has roots and.
4 Roots of quadratic equations 5 The roots of the quadratic equation x x 5 0 are and. Find a quadratic equation which has roots and. 6 The roots of the quadratic equation x 5x 7 0 are and. (a) Without solving the equation, find the values of (i), (ii) 3 3. (b) Determine an equation with integer coefficients which has roots and. [A] 7 The roots of the quadratic equation 3x 4x 0 are and. (a) Without solving the equation, find the values of (i), (ii) 3 3. (b) Determine a quadratic equation with integer coefficients which has roots 3 and 3. [A] 8 The roots of the quadratic equation x x 3 0 are and. (a) Without solving the equation: (i) write down the value of and the value of, (ii) show that 3 3 0, (iii)find the value of 3 3. (b) Determine a quadratic equation with integer coefficients which has roots 3 and. [A] 3 9 The roots of the quadratic equation x 3x 0 are and. (a) Without solving the equation: (i) show that 7, (ii) find the value of 3 3. (b) (i) Show that 4 4 ( ) (). (ii) Hence, find the value of 4 4. (c) Determine a quadratic equation with integer coefficients which has roots ( 3 ) and ( 3 ). [A] 0 The roots of the quadratic equation x 3x 0 are and. (a) Write down the values of and. (b) Without solving the equation, find the value of: (i), (ii) 3 3. (c) Determine a quadratic equation with integer coefficients 3 which has roots and. [A] 3
Roots of quadratic equations 5 Key point summary When the quadratic equation ax bx c 0 has roots and : p The sum of the roots, b a ; and the product of roots, c. a A quadratic equation can be expressed as x (sum of roots)x (product of roots) 0. p 3 Two useful results are: p5 ( ) 3 3 ( ) 3 3( ). 4 The basic method for forming new equations with roots that are related to the roots of p8 a given equation is: Write down the sum of the roots,, and the product of the roots,, of the given equation. Find the sum and product of the new roots in terms of and. 3 Write down the new equation using x (sum of new roots)x (product of new roots) 0 Test yourself Find the sums and products of the roots of the following Section. equations: (a) x 6x 4 0, (b) x x 5 0. Write down the equations, with integer coefficients, where: Section.3 (a) sum of roots 4, product of roots 7, (b) sum of roots 5 4, product of roots. What to review 3 The roots of the equation x 5x 6 0 are and. Section. Find the values of: (a) + (b). 4 Given that the roots of 3x 9x 0 are and, find Section.3 a quadratic equation whose roots are and. Test yourself ANSWERS (a) sum = 6, product = 4; (b) sum, product = 5. (a) x 4x 7 0; (b) 4x 5x 0. 3 (a) (b) 4 x 7x 7 0. 3 7 37; 3. 6