Quadratic Equations

Quadratic Equation


An equation of the form ax² + bx + c = 0, where a 0, is called a quadratic equation. The numbers a, b, c are called the coefficients of the quadratic equation. A root of the quadratic

equation is a number a (real or complex) such that a² + b + c = 0.

The roots of the given quadratic equation are given by x = .


Basic Results:

The quantity D (D = b² - 4ac) is known as the discriminant of the quadratic equation. For a, b, c real,

The quadratic equation has real and equal roots if and only if D = 0 i.e. b² - 4ac = 0.

The quadratic equation has real and distinct roots if and only if D > 0 i.e. b² - 4ac > 0.

The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 i.e. b² - 4ac < 0.

If p + iq (p and q being real) is a root of the quadratic equation where i = , then p - iq is also a root of the quadratic equation.

If p + is an irrational root of the quadratic equation, then p -is also a root of the quadratic equation provided that all the coefficients are rational.

The quadratic equation has rational roots if D is a perfect square and a, b, c are rational. If a = 1 and b, c are integers and the roots of the quadratic equation are rational, then the

roots must be integers.



Factorization Method:

In the factorization method we write the coefficient of x as a sum of two numbers l and m such that lm = ac i.e. we write

b = l + m with lm = ac

so that the equation ax² + bx + c = 0 becomes

ax² + (l + m) x + or a²x² + a (l + m) x + lm = 0

or (ax + l) (ax + m) = 0 ax + l = 0 or ax + m = 0.

Hence the roots are x = and x = . In order to obtain l and m, we write

l - m = .

With l + m = b, we get

l = and m =

so that the two roots are .

Let and be two roots of the given quadratic equation. Then + = - and = .

A quadratic equation, whose roots are a and b can be written as (x - ) (x - ) = 0

i.e., ax² + bx + c a(x - ) (x - ).


Condition for two Quadratic Equations to have a Common Root

Let ax²+bx+c = 0 and dx²+ex + f = 0 have a common root (say). Then a² + b + c = 0 and d² + e + f = 0.

Solving for ² and , we get

i.e. ² = and = (dc - af)² = (bf - ce) (ae - bd),

which is the required condition for the two equations to have a common root.



Relation between the Roots of a Polynomial Equation of Degree n Consider the equation

a_nxⁿ + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + .... + a₁x + a₀ = 0 . . . . (1)

(a₀, a₁...., a_n are real coefficients and a_n 0).

Let ₁, ₂,....,_n be the roots of equation (1). Then

a_nxⁿ + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + ..... + a₁x + a₀ a_n(x - ₁) (x - ₂) ..... (x - _n).

Comparing the coefficients of like powers of x, we get

a₁ + ₂ + ₃ + .... + _n = -,

₁a₂ + ₁₃ + ₁₄ + .... + ₂₃ + ... + _{n - 1n} = ,

₁₂ . . . . . _r + .... + _n-r+1n-r+2 ... _n = ( -1)^r ,

₁₂ ... _n = (-1)ⁿ.


Note:
A polynomial equation of degree n has n roots (real or imaginary).

If all the coefficients are real then the imaginary roots occur in pairs i.e. number of complex roots is always even.

If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.

If is a repeated root (repeating r times) of a polynomial equation f(x) = 0 of degree n i.e. f(x) = (x - )^r g(x) , where g(x) is a polynomial of degree n - r and g() 0, then f()

= f'() = f''() = . . . . = f ^(r-1)() = 0 and f ^r () 0 .

Remainder Theorem: If we divide a polynomial p(x) by x - , the remainder obtained is p(). Note that if p() = 0, then x - is a factor of p(x).

If a polynomial equation of degree n has n + 1 roots say x₁,...x_{n + 1}, (x_i x_j if i j), then the polynomial is identically zero. ie. p(x) = 0, x R.

(In other words, the coefficients a₀, .... a_n are all zero).

If p(a) and p(b) (a < b) are of opposite sign, then p(x) = 0 has odd number of roots in (a, b), i.e. it have at least one root in (a, b).

If coefficients in p(x) have 'm' changes in signs, then p(x) = 0 have at most 'm' positive real roots and if p(-x) have 't' changes in sign, then p(x) = 0 have at most 't' negative real roots.

By this we can find maximum number of real roots and minimum number of complex roots of a polynomial equations p(x) = 0.

The Method of Intervals (Wavy Curve Method)

The Method of intervals (or wavy curve method) is used for solving inequalities of the form

f(x) => 0 ( < 0, 0, or 0) where n₁, n₂,..., n_k , m₁, m₂, ...m_p are natural numbers and the numbers a₁, a₂, ... , a_k ; b₁, b₂,...b_p are any real

numbers such that a_I b_j , where i = 1, 2, 3,..., k and j = 1, 2, 3,..., p .


It consists of the following steps:

All zeros¹ of the function f(x) contained on the left hand side of the inequality should be marked on the number line with inked (black) circles.

All points of discontinuities² of the function f(x) contained on the left hand side of the inequality should be marked on the number line with un-inked (white) circles.

Check the value of f(x) for any real number greater than the right most marked number on the number line.

From right to left, beginning above the number line (in case the value of f(x) is positive in step (iii), otherwise, from below the number line), a wavy curve should be drawn to pass

through all the marked points so that when it passes through a simple point³, the curve intersects the number line, and, when passing through a double point⁴, the curve remains located

on one side of the number line.

The appropriate intervals are chosen in accordance with the sign of inequality (the function f(x) is positive whenever the curve is situated above the number line, it is negative if the

curve is found below the number line). Their union represents the solution of the given inequality.




Remark:

(i) Points of discontinuity will never be included in the answer.

(ii) If you are asked to find the intervals where f(x) is non-negative or non-positive then make the intervals closed corresponding to the roots of the numerator and let it remain open

corresponding to the roots of the denominator.

1.The point for which f(x) vanishes (becomes zero) are called function zeros e.g. x = a_i.

2. The points x = b_j are the point of the discontinuity of the function f(x).

3. If the exponents of a factor is odd then the point is called a simple point.

4. If the exponent of a factor is even then the point is called a double point.



Quadratic Expression

The expression ax² + bx + c is said to be a real quadratic expression in x where a, b, c are real and a 0. Let f(x) = ax² + bx + c where a, b, c, R (a 0).

f(x) can be rewritten as f(x) = a=a, where D = b²-4ac is the discrimnant of the quadratic expression. Then y = f(x) represents a

parabola whose axis is parallel to the y - axis, with vertex at A.

Depending on the sign of a and b² - 4ac, f(x) may be positive, negative or zero. This gives rise to the following cases:


(i) a > 0 and b²- 4ac < 0

f(x) > 0 x R.

In this case the parabola always remains above the x - axis

.


(ii) a > 0 and b² - 4ac = 0

f(x) 0 x R.

In this case the parabola touches the

x - axis at one point and remains concave upwards

.


(iii) a > 0 and b²- 4ac > 0.

Let f(x) = 0 have two real roots and (<).

Then f(x) > 0 x (-, )(, ),

and f(x) < 0 x (, )

.


(iv) a < 0 and b² - 4ac < 0

f(x) < 0 x R.

In this case the parabola always remains below the x-axis
.



(v) a < 0 and b² - 4ac = 0

f(x) 0 x R.

In this case the parabola touches the

x - axis and lies below the x - axis.





(vi) a < 0 and b²- 4ac > 0

Let f(x) = 0 have two real roots a and

( < ). Then f(x) < 0 x(-, )(, )

and f(x) > 0 x(, )
.



Interval in Which the Roots Lie

In some problems we want the roots of the equation ax² + bx + c = 0 to lie in a given interval. For this we impose conditions on a, b and c. Let f(x) = ax² + bx + c.

If both the roots are positive i.e. they lie in (0, ), then the sum of the roots as well as the product of the roots must be positive

+ = - and = with ² - 4ac 0.

Similarly, if both the roots are negative i.e. they lie in (- , 0) then the sum of the roots will be negative and the product of the roots must be positive

i.e. + = -< 0 and = with ² - 4ac 0.

Both the roots are greater than a given number k if the following three conditions are satisfied: D 0, -and a.f(k) > 0.

Both the roots will be less than a given number k if the following conditions are satisfied: D 0, -< k and a.f(k) > 0.

Both the roots will lie in the given interval (k₁, k₂) if the following conditions are satisfied: D 0 k₁ < - and a. f(k₁) > 0, a.f(k₂) > 0.

Exactly one of the roots lies in the given interval (k₁, k₂) if f(k₁) . f(k₂) < 0.

A given number k will lie between the roots if a.f(k) < 0.

In particular, the roots of the equation will be of opposite signs if 0 lies between the roots a.f(0) < 0. It also implies that the product of the roots is negative.


Cube Root of Unity:

Consider the equation x³ = 1 x³ - 1 = 0. We note that x = 1 is one of the roots of this equation, so that

0 = x³ - 1 = (x - 1) (x² + x + 1)

x = 1 or x² + x + 1 = 0 x =.

Hence the cube roots of unity are: x = 1, x = - +, x = - -.

Alternatively, x = 1, , ² where =.