Logo

Quadratic Equations

Polynomial

f(x) = a0 + a1x + a2x2 + a3x3+ ... +an–1xn–1 + anxn, where x is a variable, a > 0, (<0) are constants and an ≠ 0.

Example: 4x4 + 3x3 – 7x2 + 5x + 3, 3x3 + x2 – 3x + 5.

(1)    Real polynomial

f(x) = a0 + a1x + a2x2 + a3x3 + ... +anxn

is called real polynomial of real variable x with real coefficients.

Example:  3x3 – 4x2 + 5x – 4, x2 – 2x + 1 etc. are real polynomials.

(2)   Complex polynomial:

f(x) = a0 + a1x + a2x2 + a3x3 + ... + anxn

is called complex polynomial of complex variable x with complex coefficients.

Example: 3x2 – (2 + 4i)x + (5i – 4), x3 – 4ix2 + (1 + 2i)x + 4 etc. are complex polynomials.

(3)   Degree of polynomial:

Highest power of variable x in a polynomial is called degree of polynomial.

Example: f(x) = a0 + a1x + a2x2 + ... an–1xn–1 + anxn is a n degree polynomial.

f(x) = 4x3 + 3x2 – 7x + 5 is a 3 degree polynomial.

(4)   Polynomial equation:

If f(x) is a polynomial, real or complex, then f(x) = 0 is called a polynomial equation.

 

Roots of a quadratic equation:

The values of variable x which satisfy the quadratic equation is called roots of quadratic equation.

 

Solution of Quadratic Equation

(1)    Factorization method:

Let ax2 + bx + c = a(x – α)(x – β) = 0.

          Then x = α and

          x = β will satisfy the given equation.

Hence, factorize the equation and equating each factor to zero gives roots of the equation.

Example: 3x2 – 2x + 1 = 0 ⇒ (x – 1) (3x + 1) = 0;  x = 1, –1/3

(2)   Sri Dharacharya method:

The quadratic equation ax2 + bx + c = 0 (a 0) has two roots, given by

Sri Dharacharya method1 Sri Dharacharya method

Every quadratic equation has two and only two roots.

 

Nature of roots:

In a quadratic equation ax2 + bx + c = 0, let us suppose that a, b, c are real and a 0. The following is true about the nature of its roots.

  1. The equation has real and distinct roots if and only if D ≡ b2 – 4ac > 0.

  2. The equation has real and coincident (equal) roots if and only if D ≡ b2 – 4ac = 0.

  3. The equation has complex roots of the form α ± iβ, a 0, β 0 ∈ R  if and only if D ≡ b2 – 4ac < 0.

  4. The equation has rational roots if and only if a, b, c ∈ Q (the set of rational numbers) and D ≡ b2 – 4ac is a perfect square (of a rational number).

  5. The equation has (unequal) irrational (surd form) roots if and only if D ≡ b2 – 4ac > 0 and not a perfect square even if a, b and c are rational. In this case if p + √q, p,q rational is an irrational root, then p – √q is also a root (a, b, c being rational).

  6. α + iβ (β ≠ 0 and α, β Î R) is a root if and only if its conjugate α – iβ is a root, that is complex roots occur in pairs in a quadratic equation. In case the equation is satisfied by more than two complex numbers, then it reduces to an identity. 0.x2 + 0.x + 0 = 0, i.e. a = 0 = b = c.

 

Relations between roots and coefficients

Relation between roots and coefficients of quadratic equation:

If a and b are the roots of quadratic equation ax2 + ax2 + bx + c = 0, (α ≠ 0) then

Sum of roots = S = α + β = (-a/b)

Sum of roots

Product of roots = P = α.β = (c/a)

Product of roots

Formation of an equation with given roots:

A quadratic equation whose roots are a and b is given by (x – α) (x – β) = 0.

∴ x2 – (α + β) + αβ = 0

i.e. x2 – (sum of roots)x + (product of roots) = 0

∴ x2 – Sx + P = 0

Symmetric function of the roots:

A function of α and β is said to be a symmetric function, if it remains unchanged when α and β are interchanged.

For example, α2 + β2 + 2 αβ is a symmetric function of α and β whereas α2 – β2 + 3 αβ is not a symmetric function of  α and β.

In order to find the value of a symmetric function of α and β, express the given function in terms of α + β and αβ. The following results may be useful.

  1. α2 + β = (α + β)2 – 2αβ

  2. α3 + β3 = (α + β)3 – 3αβ(α + β)

  3. α4 + β4 = (α3 + β3) (α + β) – αβ(α2 + β2)

  4. α5 + β5 = (α3 + β3) (α2 + β2) – α2β2(α + β)

  5. |α–β| = √((α + β)2 – 4αβ)

  6. α2β2 = (α+β)( α–β)

  7. α3 – β3 = (α – β)[(α + β)2 – αβ]

  8. α4 – β4 = (α + β) (α – β) (α2 + β2)

 

Higher Degree Equations

The equation p(x) = a0xn + a1xn–1 + ... + an–1x + an = 0 ..... (1)

Where the coefficients a0, a1, ... an e R (or C) and a0 ≠ 0 is called an equation of nth degree, which has exactly n roots α1¸ α2, ... αn ∈ C,  then we can write

p(x) = a0 (x – α1) (x – α2) ... (x – αn)

∑α1 = α1 + α2 + ... + αn = –(a1/a0)

∑α1α2 = α1α2 + ... + αn–1 αn = (a2/a0) and so on and

α1α2 ... αn = (–1)n(an/a­0)

 

Cubic equation:

When n = 3, the equation is a cubic of the form ax3 + bx2 + cx + d = 0, and we have in this case α + β + γ = –(a/b); αβ + β γ + γα = (c/a); αβγ = –(d/a)

 

Biquadratic equation:

If α,β,γ,δ are roots of the biquadratic equation ax4 + bx3 + cx2 + dx + e = 0, then

σ2 = αβ + αγ + αδ + βγ + βδ + γδ = c/a

σ4 = αβγδ = e/a

                  

Formation of a Polynomial Equation from Given Roots

If α1, α2, α3, ... αn are the roots of a polynomial equation of degree n, then the equation is xn – σn – σ1xn–1 + σ2xn–1 – σ3xn–3+...+ (–1)n σn = 0 where σr = ∑α1 α2 ... αr

 

Cubic equation:

If α,β,γ are the roots of a cubic equation, then the equation is x3 – σ1x2 + σ2x – σ3 = 0 or x3 – (α+β+γ)x2 + (αβ + αγ + βγ)x – αβγ = 0.

 

Biquadratic Equation:

If α,β,γ,δ are the roots of a biquadratic equation, then the equation is x4 – σ1x3 + σ2x2 – σ3x + σ4 = 0 or x4 – (α + β + γ + δ)x3 + (αβ + αγ + αδ + βδ + γδ)x2 – (αβγ + αβδ + αγδ + βγδ)x + αβγδ = 0

 

Condition for Common Roots

(1)     Only one root is common:

Let a be the common root of quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0.

∴ a1α2 + b1α + c1 = 0, a2α2 + b2α + c2 = 0

By Crammer’s rule:

Crammer’s rule

The condition for only one root common is

(c1a2 – c2a1)2 = (b1c2 – b2c1) (a1b2 – a2b1)

(2)     Both roots are common:

Then required condition is (a1/a2) = (b1/b2) = (c1/c­2)

 

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 f(x) = 0 lie between a and b.

  2. If f(a) = f(b) then there exists a point c between a and b such that f'(c) = 0, a < c < b.

  3. If a is a root of the equation f(x) = 0 then the polynomial f(x) is exactly divisible by (x – α), then (x – α) is factor of f(x).

  4. If the roots of the quadratic equations a2x2 + b1x + c1 = 0 and a2x2 + b2x2 + c2 = 0 are in the same ratio (i.e. (α11) = (α22)) then b12/b12 = a1c1/a2c2.

 

The Quadratic Expression

(1)    Let f(x) ≡ ax2 + bx + c,  a, b,  c ∈ R, a > 0 be a quadratic expression. Since,

The Quadratic Expression  ……(i)

The following is true from equation (i)

(i)     f(x) > 0 (<0) for all values of x ∈  R if and only if a > 0 (<0)  and D ≡  b2 – 4ac < 0..

(ii)    f(x) ≥  0 (≤0) if and only if a > 0(<0) and D ≡  b2 – 4ac = 0.

In this case (D = 0), f(x) = 0 if and only if x = –(b/2a)

(iii)   If D ≡  b2 – 4ac  > 0  and a > 0(<0), then

Quadratic Expression

(iv)   If a > 0, (<0) then f(x) has a minimum (maximum) value at x = –(b/2a) and this value is given by [f(x)]min(max) = (4ac-b2)/4a.

(2)   Sign of quadratic expression:

Let f(x) = ax2 + bx + c or y = ax2 + bx + c

Where a, b, cR 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 f(x) > 0 for all x ∈ R i.e., f(x) is positive for all real values of x.

(ii)   a < 0 and D < 0, so f(x) < 0 for all xR i.e., f(x) is negative for all real values of x.

(iii)  a > 0 and D = 0, so f(x) ≥ 0 for all xR i.e., f(x) is positive for all real values of x except at vertex, where f(x) = 0.

(iv)  a < 0 and D = 0, so f(x) ≤ 0 for all xR i.e. f(x) is negative for all real values of x except at vertex, where f(x) = 0.

(v)    a > 0 and D > 0,  let f(x) = 0 have two real roots α and β (α < β), then f(x) > 0 for all x ∈ (–∞,α) ⋃ (β, ∞) and f(x) < 0 for all x ∈(α,β).

(vi)  a < 0 and D > 0, let f(x) = 0 have two real roots α and β (α < β). Then f(x) < 0 for all x ∈ (–∞,α) ⋃ (β, ∞) and f(x) > 0 for all x ∈(α,β)

(3)   Graph of a quadratic expression:

y = ax2 + bx + c = f(x)

(i)     The graph of the curve y = f(x) is parabolic.

(ii)   The axis of parabola is X = 0 or x + (b/2a) = 0

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.

Graph of a quadratic expression

(iv)  Intersection with axis

(a)   Intersection with x-axis: For x axis, y = 0 ⇒ ax2 +bx + c = ⇒ x = (–b ± √D)/2a

For D > 0, parabola cuts x-axis in two real and distinct points i.e. x = (–b ± √D)/2a

For D=0, parabola touches x-axis in one point, x = –b/2a.

Intersection with x-axis

For D < 0, parabola does not cut x-axis (i.e. imaginary value of x).

parabola

(b)   Intersection with axis y-axis: For y axis x = 0, y = c.

 

Position of Roots

  1. If f(x) = 0 is an equation and a, b are two real numbers such that f(a).f(b) < 0 has at least one real root or an odd number of real roots between a and b. In case f(a) and f(b) are of the same sign, then either no real root or an even number of real roots of f(x) = 0 lie between a and b.

  2. Every equation of an odd degree has at least one real root, whose sign is opposite to that of its last term, provided the coefficient of the first term is +ve e.g., x3 – 3x + 2 = 0 has one real negative root.

  3. Every equation of an even degree whose last term is –ve  and the coefficient of first term +ve has at least two real roots, one +ve and one –ve e.g.,  x4 + 4x3 + 3x2 + 5x – 2 = 0 has at least two real roots, one +ve and one –ve.

  4. If an equation has only one change of sign, it has one +ve root and no more.

  5. If all the terms of an equation are +ve and the equation involves no odd power of x, then all its roots are complex.

 

Descarte's Rule of Signs

The maximum number of positive real roots of a polynomial equation f(x) = 0 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 f(x) = 0 is the number of changes of sign from positive to negative and negative to positive in f(–x).

 

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 (a1x2 + b1x + c1)/ (a2x2 + b2x + c2) for real values of x, the following algorithm is used

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

Solution of rational algebraic inequation

are known as rational algebraic inequations.

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 P(x)/Q(x) bears positive sign and in other regions the expression bears positive and negative signs depending on the exponents of the factors.

(3)   Lagrange’s identity

If a1,a2,a3,b1,b2,b3 ∈ R then 

(a12 + a22 + a32) (b12 b22 + b32) – (a1b1 + a2b2 + a3b3)2

= (a1b2 – a2b1)2 + (a2b3 – a­3b2)2 + (a3b1 – a1b3)2

 

Equations which can be reduced to Linear, Quadratic and Biquadratic Equations

Type I : An equation of the form (x – a) (x – b) (x – c) (x – d) = A,

where  a < b < c < d, b – a = d – c, can be solved by a change of variable.

i.e.,

Biquadratic Equations

Quadratic and Biquadratic Equations

Type II: An equation of the form

(x – a) (x – b) (x – c) (x – d) = Ax2

where ab = cd, can be reduced to a collection of two quadratic equations by a change of variable y = x + (ab/x)

Type III: An equation of the form (x – a)4 + (x – b)4 = A can also be solved by a change of variable, i.e., making a substitution y = ((x – a) + (x – b))/2.

 

Some Important Result:

(1)    For the quadratic equation ax2 + bx + c = 0.

(i)       One root will be reciprocal of the other if a = c.

(ii)      One root is zero if c = 0.

(iii)    Roots are equal in magnitude but opposite in sign if b = 0.

(iv)     Both roots are zero if b = c = 0.

(v)      Roots are positive if a and c are of the same sign and b is of the opposite sign.

(vi)     Roots are of opposite sign if a and c are of opposite sign.

(vii)   Roots are negative if a,b,c are of the same sign.

(2)   Let f(x) = ax2 + bx + c, where a > 0. Then

(i)      Conditions for both the roots of f(x) = 0 to be greater than a given number k are b2 – 4ac ≥ 0; f(k) = 0; (–b/2a) > k.

(ii)     Conditions for both the roots of f(x) = 0 to be less than a given number k are b2 – 4ac ≥0, f(k) > 0, (–b/2a) < k.

(iii)   The number k lies between the roots of f(x) = 0, if b2 – 4ac > 0; f(k) < 0.

(iv)     Conditions for exactly one root of f(x) = 0 to lie between k1 and k2 is f(k1) f(k2) < 0, b2 – 4ac > 0.

(v)     Conditions for both the roots of f(x) = 0 are confined between k1 and k2 is f(k1) > 0, f(k2) > 0, b2 – 4ac ≥ 0 and k1 < (–b/2a) < k2, where k1 < k2.

(vi)    Conditions for both the numbers k1 and k2 lie between the roots of f(x) = 0 is b2 – 4ac > 0; f(k1) < 0; f(k2) < 0.

 

Points at the Glance

An equation of degree n has n roots, real or imaginary.

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

If f(α) = 0 and f'(α) = 0, then α is a repeated root of the quadratic equation f(x) = 0 and f(x) = a(x – α)2. In fact α = –b/2a.

If α is a repeated root of the quadratic equation f(x) = ax2 + bx + c = 0

Then α is also a root of the equation f'(x) = 0

If α is repeated common root of two quadratic equations f(x) = 0 and ∅(x) = 0, then α is also a common root of the equations f'(x) = 0 and ∅'(x) = 0.

In the equation ax2 + bx + c = 0 [a, b, c ∈ R], if a + b + c = 0 then the roots are 1, c/a  and if a – b + c = 0, then the roots are –1 and – c/a.           

If the ratio of roots of the quadratic equation ax2 + bx + c = 0 be p : q, then pqb2 = (p + q)2 ac .

If one root of the quadratic equation ax2 + bx + c = 0 is equal to the nth power of the other, then  quadratic equation.

If the roots of the equation ax2 + bc + c = 0 are α, β, then the roots of cx2 + bx + a = 0 will be 1/α, 1/β.

The roots of the equation ax2 + bx + c = 0 are reciprocal to a'x2 + b'x + c' = 0 if (cc' –aa')2 = (ba' –cb') (ab' – bc').

If one root is k times the other root of the quadratic equation ax2 + bx + c = 0, then (k + 1)2/k = b2/ac.

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 its all roots are complex.

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.