Roots Of Quadratic Equations

Roots of quadratic equations If an algebraic equation, in which the unknown quantity is x, is satisfied by putting x = c, we say that c is a root of the equation. For example x2 -5x + 6 = 0 is satisfied by putting x = 2, so one root of this equation is 2 (the other is 3). It is often useful to be able to obtain information about the roots of an equation without actually solving it. For instance, if α and β are the roots of the equation 3x2 + x -1 = 0, the value of α2+β2can be found without first finding the values of α and β. This is done by finding the values of α+β and αβ, and expressing α2+β2in terms of α+β and αβ. The equation whose roots are α and β may be written ( x −α )( x − β ) = 0 ∴ x − αx − βx + αβ = 0 2

(1)

∴ x 2 − (α + β ) x + αβ = 0

But suppose that α and β are also the roots of the equation ax2 + bx + c = 0 which may be written x2 +

b c x+ =0 a a

(2 )

Now equations (1) and (2), having the same roots, must be precisely the same equation, written in two different ways, since the coefficients of x2 are both 1. Therefore (a) the coefficients of x must be equal, b .'. α + β = − a

(b) the constant terms must be equal, ∴αβ =

c a

Note. If it is required to write down an equation whose roots are known, equation (1) gives it in a convenient form. It may be written: x2 — (sum of the roots)x + (product of the roots) = 0 Write down the sums and products of the roots of the following equations: (a) 3x2 - 2x - 7 = 0, (c) 2x2 + 5x = 1,

(b) 5x2 + llx + 3 = 0, (d) 2x(x + 1) = x + 7.

Write down equations, the sums and products of whose roots are respectively: (a) 7, 12;

(b) 3, -2;

(c)

−

1 3 ,− 2 8

2

(d) 3 ,0 .

Write down the sum and product of the roots of the equation 3x2 + 9x + 7 = 0. Example 1 The roots of the equation 3x2 + 4x - 5 = 0 are α, β. Find the values of (a) 1

α

+

1

β

Both

, (b) α2 + β2. 1

α

+

1

β

4 α +β =− , 3

and α2 + β2 can be expressed in terms of α + β and αβ.

αβ = −

4 1 1 α +β 4 + = = 3 = . 5 α β αβ 5 − 3 −

(a)

5 3

(b)

α 2 + β 2 = α 2 + 2α β + β 2 − 2α β 2

 4  5 2 = ( α + β ) − 2α β=  −  − 2 −  3    3

∴α 2 + β 2 =

16 10 46 + = 9 3 9

Alternatively, since α and β are roots of the equation 3x2 + 4x — 5 = 0,

3α 2 + 4α − 5 = 0

3β 2 + 4 β − 5 = 0

Adding,

(

)

3 α 2 + β 2 + 4( α + β ) − 10 = 0

(

)

16 − 10 = 0 3 16 10 46 ∴α 2 + β 2 = + = 9 3 9

∴3 α 2 + β 2 −

Example 2 The roots of the equation 2x2 - 7x + 4 = 0 are α,β Find an equation

with integral coefficients whose roots are

α β , . β α

Since α, β are the roots of the equation 2x2 -7x + 4 = 0, we have α +β =

7 , αβ = 2 2

Then the required equation may be formed from equation (1) above, if the sum and

product of

α β , are expressed in terms of α + β and αβ . β α

α β α 2 + β 2 ( α + β ) − 2α β + = = β α α β α β 49 −4 33 = 4 = 2 8 2

Therefore the sum of the roots is

33 8

.

α β × =1 β α

Therefore the product of the roots is 1. Hence the equation with roots x2 −

α β , is β α

33 x +1 = 0 8

Multiplying through by 8, in order to obtain integral coefficients, the required equation is 8x2 - 33x + 8 = 0 Symmetrical functions The functions of α and β that have been used in this chapter all show a certain symmetry. Consider, for example, α + β , αβ ,

1

α

+

1

β

, α2 + β 2 ,

α β + β α

Notice that if α and β are interchanged: β +α , βα,

1

β

+

1

α

, β 2 +α 2 ,

β α + α β

the resulting functions are the same. When a function of α and β is unchanged when α and β are interchanged, it is called a symmetrical function of α and β. Such functions occurring in this chapter may be expressed in terms of α + β and αβ, as in the next example. Example Express in terms of α + β and αβ : (a) α3 + β3, (b) (α -β)2.

(a) α3 and β3 occur in the expansion of (α+ β)3. (α + β)3 = α3 + 3α2β + 3αβ2 + β3 .'. α3 + β3 = (α + β)3 - 3α2β - 3αβ2 .'. α3 + β3 = (α + β)3 - 3αβ (α +β) (b) (α - β)2 = α2 - 2αβ + β2. α2 and β2 occur in the expansion of (α β) + 2. (α + β)2 = α2 + 2αβ + β2 (α - β)2=(α + β)2 -4αβ

Exercise 1 Find the sums and products of the roots of the following equations: (a) 2 x 2 - l l x + 3 = 0, (d)x 2 + x = l , (g)

x+

(b) 2x2 + x - l = 0 , (e) t(t-l) = 3,

1 =4 x

(h)

(c) 3x2 = 7x + 6, (f) y(y+1)=2y+5

1 1 1 + = t t +1 2

2 Find equations, with integral coefficients, the sums and products of the equations whose roots are respectively: (a) 3, 4;

3 2

(c) , −

(b) -5, 6;

(e) 0, -7;

(f) 1.2,0.8;

5 ; 2

(d) −

1 1 ; 3 36

(g) − ,

7 , 0; 3 (h) -2.5, -1.6.

3 The roots of the equation 2 x2 + 3x - 4 = 0 are α, β. Find the values of (a) (b)

1

α

+

1

β

,

(c) (α+ 1) (β + 1),

(d)

β α + α β.

α2 + β 2,

4 If the roots of the equation 3 x2 — 5x + 1 = 0 are α, β, find the values of 2

2

(a) αβ + α β,

2

2

(b) α -αβ + β ,

3

3

(c) α + β ,

(d)

α2 β 2 + β α

.

5 The equation 4x2 + 8x -1 = 0 has roots α, β. Find the values of (a)

1

α

2

+

1

β

2

,

(b) (α - β)2,

(c) α3β + αβ3,

(d)

1

α β 2

+

1

α β2

.

6 If the roots of the equation x2 — 5x — 7 = 0 are α, β. Find equations whose roots are (a) α2, β2;

(b) α+1,β+1;

(c) α2β,αβ2.

7 The roots of the equation 2x2 — 4x + 1 = 0 are α, β. Find equations with integral coefficients whose roots are (a) α - 2, β - 2;

1

1

(b) α , β ;

α β

(c) β , α

8 Find an equation, with integral coefficients, whose roots are the squares of the roots of the equation 2x2 + 5x — 6 = 0. 9 The roots of the equation x2 + 6x + q = 0 are α and β -1. Find the value of q. 10 The roots of the equation x2 — px + 8 = 0 are α and α+ 2. Find two possible values of p. 11 The roots of the equationx2 + 2px + q = 0 differ by 2. Show thatp2 = 1 + q. 12 If the roots of the equationax2 + bx + c = 0 are α, β, find expressions in terms of a, b, c for (a) α2β + αβ2, (d)

1

α

+

1

(b) α2 + β2,(c) α3 + β3, (e)

β

α β + β α,

(f) α4 + β4

13 The equation ax2 + bx + c = 0 has roots α, β. Find equations whose rootsare (a) -α,-β (d)

−

1

α

−

(b) α + 1 ,β + 1 ; 1

β

(e) α – β, β - α;

(c) α2, β2; (f) 2α + β, α + 2β.

14 Prove that, if the difference between the roots of the equation

ax2 + bx + c = 0 is 1, then a2 = b2 - 4ac. 15 Prove that, if one root of the equation xa2 + bx + c = 0 is twice the other, then 2b2 = 9ac. 16 Prove that, if the sum of the squares of the roots of the equation ax2 + bx + c = 0 is 1, then b2 = 2ac + a2. 17 Prove that, if the sum of the reciprocals of the roots of the equation ax2 + bx + c = 0 is 1, then b + c = 0. If, in addition, one root of the equation is twice the other, use the result of No. 15 to find one set of values of a, b, c. Solve the equation. 18 In the equation ax2 + bx + c = 0, make the substitutions (b) x = y2,

(a) x = y - 1,

(c) x =

y

,

and simplify the equations. If the roots of the equation xa2 + bx + c = 0 are α, β, what are the roots of the three equations iny? [Express y in terms of x, and give your answers in terms of α,β.] 19 If the roots of the equation ax2 + bx + c = 0 are α, β, make substitutions, as in No. 18, to find equations whose roots are (a) α + 2, β + 2;

1± α , 1± β

.

1

1

(b) α , β ;

(c)