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Maths Class 11 Chapter 5 Part -1 Quadratic equations
1. Real Polynomial: Let a0, a1, a2, … , an be real numbers and x is a real variable. Then, f(x) =
a0 + a1x + a2x2 + … + anxn is called a real polynomial of real variable x with real coefficients.
2. Complex Polynomial: If a0, a1, a2, … , an be complex numbers and x is a varying complex
number, then f(x) = a0 + a1x + a2x2 + … + an – 1xn – 1 + anxn is called a complex polynomial or a
polynomial of complex variable with complex coefficients.
3. Degree of a Polynomial: A polynomial f(x) = a0 + a1x + a2x2 + a3x3 + … + anxn , real or
complex is a polynomial of degree n , if an ≠ 0.
4. Polynomial Equation: If f(x) is a polynomial, real or complex, then f(x) = 0 is called a
polynomial equation. If f(x) is a polynomial of second degree, then f(x) = 0 is called a
quadratic equation .
Quadratic Equation: A polynomial of second degree is called a quadratic polynomial.
Polynomials of degree three and four are known as cubic and biquadratic polynomials
respectively. A quadratic polynomial f(x) when equated to zero is called quadratic equation.
i.e., ax2 + bx + c = 0 where a ≠ 0.
Roots of a Quadratic Equation: The values of variable x .which satisfy the quadratic equation
is called roots of quadratic equation.
Important Points to be Remembered
An equation of degree n has n roots, real or imaginary .
Surd and imaginary roots always occur in pairs of a polynomial equation with real
coefficients i.e., if (√2 + √3i) is a root of an equation, then’ (√2 – √3i) 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.
If an equation has only one change of sign it has one positive root.
If all the terms of an equation are positive and the equation involves odd powers of x,
then all its roots are complex.
Solution of Quadratic Equation
1.Factorization Method: Let ax2 + bx + c = α(x – α) (x – β) = O. Then, x = α and x = β will
satisfy the given equation.
2. Direct Formula: Quadratic equation ax2 + bx + c = 0 (a ≠ 0) has two roots, given by
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where D = Δ = b2 – 4ac is called discriminant of the equation .
Above formulas also known as Sridharacharya formula.
Nature of Roots
Let quadratic equation be ax2 + bx + c = 0, whose discriminant is D.
(i) For ax2 + bx + c = 0; a, b , C ∈ R and a ≠ 0, if
(a) D < => Complex roots
(b) D > 0 => Real and distinct roots
(c) D = 0 => Real and equal roots as α = β = – b/2a
(ii) If a, b, C ∈ Q, a ≠ 0, then
(a) If D > 0 and D is a perfect square => Roots are unequal and rational.
(b) If D > 0, a = 1; b, c ∈ I and D is a perfect square. => Roots are integral. .
(c) If D > and D is not a perfect square. => Roots are irrational and unequal.
(iii) Conjugate Roots The irrational and complex roots of a quadratic equation always occur in
pairs. Therefore,
(a) If one root be α + iβ, then other root will be α – iβ.
(b) If one root be α + √β, then other root will be α – √β.
(iv) If D, and D2 be the discriminants of two quadratic equations, then
(a) If D1 + D2 ≥ 0, then At least one of D1 and D2 ≥ 0 If D1 < 0, then D2 > 0 ,
(b) If D1 + D2 < 0, then At least one of D1 and D2 < 0 If D1 > 0, then D2 < 0
Roots Under Particular Conditions
For the quadratic equation ax2 + bx + e = 0.
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(i) If b = 0 => Roots are real/complex as (c < 0/c > 0) and equal in magnitude but of opposite
sign.
(ii) If c = 0 => One roots is zero, other is – b / a.
(iii) If b = C = 0 => Both roots are zero.
(iv) If a = c => Roots are reciprocal to each other.
(v) If a > 0, c < 0, a < 0, c > 0} => Roots are of opposite sign.
(vi) If a > 0, b > 0, c > 0, a < 0, b < 0, c < 0} => Both roots are negative, provided D ≥ 0
(vii) If a > 0, b < 0, c > 0, a < 0, b > 0, c < 0} => Both roots are positive, provided D ≥ 0
(viii) If sign of a = sign of b ≠ sign of c => Greater root in magnitude is negative.
(ix) If sign of b = sign of c ≠. sign of a => Greater root in magnitude is positive.
(x) If a + b + c = 0 => One root is 1 and second root is c/a.
Relation between Roots and Coefficients
1. Quadratic Equation: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
Sum of roost = S = α + β = -b/a = – coefficient of x / coefficient of x2 Product of roots = P = α
* β = c/a = constant term / coefficient of x2
2. Cubic Equation: If α, β and γ are the roots of cubic equation ax3 + bx2 + cx + d = 0.
Then,
3. Biquadratic Equation: If α, β, γ and δ are the roots of the biquadratic equation ax4 + bx3 +
cx2 + dx + e = 0, then
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Symmetric Roots: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
Formation of Polynomial Equation from Given Roots
If a1, a2 a3,…, an are the roots of an nth degree equation, then the equation is xn – S1Xn – 1 +
S2Xn – 2 – S3Xn – 3 +…+( _l)n Sn = 0 where Sn denotes the sum of the products of roots taken n at
a time.
1. Quadratic Equation
Maths Class 11 Chapter 5 Part -1 Quadratic equations
1. Real Polynomial: Let a0, a1, a2, … , an be real numbers and x is a real variable. Then, f(x) =
a0 + a1x + a2x2 + … + anxn is called a real polynomial of real variable x with real coefficients.
2. Complex Polynomial: If a0, a1, a2, … , an be complex numbers and x is a varying complex
number, then f(x) = a0 + a1x + a2x2 + … + an – 1xn – 1 + anxn is called a complex polynomial or a
polynomial of complex variable with complex coefficients.
3. Degree of a Polynomial: A polynomial f(x) = a0 + a1x + a2x2 + a3x3 + … + anxn , real or
complex is a polynomial of degree n , if an ≠ 0.
4. Polynomial Equation: If f(x) is a polynomial, real or complex, then f(x) = 0 is called a
polynomial equation. If f(x) is a polynomial of second degree, then f(x) = 0 is called a
quadratic equation .
Quadratic Equation: A polynomial of second degree is called a quadratic polynomial.
Polynomials of degree three and four are known as cubic and biquadratic polynomials
respectively. A quadratic polynomial f(x) when equated to zero is called quadratic equation.
i.e., ax2 + bx + c = 0 where a ≠ 0.
Roots of a Quadratic Equation: The values of variable x .which satisfy the quadratic equation
is called roots of quadratic equation.
Important Points to be Remembered
An equation of degree n has n roots, real or imaginary .
Surd and imaginary roots always occur in pairs of a polynomial equation with real
coefficients i.e., if (√2 + √3i) is a root of an equation, then’ (√2 – √3i) 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.
If an equation has only one change of sign it has one positive root.
If all the terms of an equation are positive and the equation involves odd powers of x,
then all its roots are complex.
Solution of Quadratic Equation
1.Factorization Method: Let ax2 + bx + c = α(x – α) (x – β) = O. Then, x = α and x = β will
satisfy the given equation.
2. Direct Formula: Quadratic equation ax2 + bx + c = 0 (a ≠ 0) has two roots, given by
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where D = Δ = b2 – 4ac is called discriminant of the equation .
Above formulas also known as Sridharacharya formula.
Nature of Roots
Let quadratic equation be ax2 + bx + c = 0, whose discriminant is D.
(i) For ax2 + bx + c = 0; a, b , C ∈ R and a ≠ 0, if
(a) D < => Complex roots
(b) D > 0 => Real and distinct roots
(c) D = 0 => Real and equal roots as α = β = – b/2a
(ii) If a, b, C ∈ Q, a ≠ 0, then
(a) If D > 0 and D is a perfect square => Roots are unequal and rational.
(b) If D > 0, a = 1; b, c ∈ I and D is a perfect square. => Roots are integral. .
(c) If D > and D is not a perfect square. => Roots are irrational and unequal.
(iii) Conjugate Roots The irrational and complex roots of a quadratic equation always occur in
pairs. Therefore,
(a) If one root be α + iβ, then other root will be α – iβ.
(b) If one root be α + √β, then other root will be α – √β.
(iv) If D, and D2 be the discriminants of two quadratic equations, then
(a) If D1 + D2 ≥ 0, then At least one of D1 and D2 ≥ 0 If D1 < 0, then D2 > 0 ,
(b) If D1 + D2 < 0, then At least one of D1 and D2 < 0 If D1 > 0, then D2 < 0
Roots Under Particular Conditions
For the quadratic equation ax2 + bx + e = 0.
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(i) If b = 0 => Roots are real/complex as (c < 0/c > 0) and equal in magnitude but of opposite
sign.
(ii) If c = 0 => One roots is zero, other is – b / a.
(iii) If b = C = 0 => Both roots are zero.
(iv) If a = c => Roots are reciprocal to each other.
(v) If a > 0, c < 0, a < 0, c > 0} => Roots are of opposite sign.
(vi) If a > 0, b > 0, c > 0, a < 0, b < 0, c < 0} => Both roots are negative, provided D ≥ 0
(vii) If a > 0, b < 0, c > 0, a < 0, b > 0, c < 0} => Both roots are positive, provided D ≥ 0
(viii) If sign of a = sign of b ≠ sign of c => Greater root in magnitude is negative.
(ix) If sign of b = sign of c ≠. sign of a => Greater root in magnitude is positive.
(x) If a + b + c = 0 => One root is 1 and second root is c/a.
Relation between Roots and Coefficients
1. Quadratic Equation: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
Sum of roost = S = α + β = -b/a = – coefficient of x / coefficient of x2 Product of roots = P = α
* β = c/a = constant term / coefficient of x2
2. Cubic Equation: If α, β and γ are the roots of cubic equation ax3 + bx2 + cx + d = 0.
Then,
3. Biquadratic Equation: If α, β, γ and δ are the roots of the biquadratic equation ax4 + bx3 +
cx2 + dx + e = 0, then
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Symmetric Roots: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
Formation of Polynomial Equation from Given Roots
If a1, a2 a3,…, an are the roots of an nth degree equation, then the equation is xn – S1Xn – 1 +
S2Xn – 2 – S3Xn – 3 +…+( _l)n Sn = 0 where Sn denotes the sum of the products of roots taken n at
a time.
1. Quadratic Equation
