Course:-Mathematics
Topic:- QUADRATIC EQUATIONS
Definition:- When we equate quadratic polynomial of the form (𝑎𝑥 2 + 𝑏𝑥 + 𝑐) equal to
zero we get a quadratic equation, where a, b and c are real numbers and a≠ 0.
Polynomial and Polynomial Equation:
As an expression of the form 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−2 +. . . +𝑎𝑛−1 𝑥 + 𝑎𝑛 where 𝑎0 , 𝑎1 , 𝑎2 … 𝑎𝑛 are
constant (𝑎 ≠ 0) and n is a positive integer, then it is called a polynomial in 𝑥 of degree 𝑛.
If 𝑓(𝑥) is a real or complex polynomial, then 𝑓(𝑥) = 0 is known as a polynomial equation.
Eg. If 𝑥 2 + 3𝑥 + 2 is a real polynomial, then 𝑥 2 + 3𝑥 + 2 = 0 is a polynomial equation.
Quadratic Equation:
If 𝑓(𝑥) is a polynomial of degree 2, then 𝑓(𝑥) = 0 is called a quadratic equation. The
general form of a quadratic equation is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏 and 𝑐 are real
numbers and 𝑎 ≠ 0. Here, 𝑥 is the variable and 𝑎, 𝑏, 𝑐 are the real coefficients.
Roots of a Quadratic Equation:
The values of the variable satisfying the given quadratic equations are called roots of that
equation. In other words, 𝑥 = 𝛼 is a root of the equation, 𝑓(𝑥) = 0, if 𝑓(𝛼) = 0.
The set of all roots of an equation, in a given domain is called the solution set of the
equation. The quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, 𝑐 ∈ ℝ and 𝑎 ≠ 0 has two
roots, namely
−𝑏+√𝐷 −𝑏−√𝐷
𝛼= and 𝛽 =
2𝑎 2𝑎
where, 𝐷 = 𝑏 2 − 4𝑎𝑐 is called the discriminant.
, Example 1. If 𝑥 2 + 𝑥 − 6 is a factor of polynomial 𝑥 3 + 𝑃𝑥 2 + 𝑄, then what are the values
of 𝑃 and 𝑄?
Soln: Let 𝑃(𝑥) = 𝑥 3 + 𝑃𝑥 2 + 𝑄
Since 𝑥 2 + 𝑥 − 6 is a factor of 𝑃(𝑥).
∴ Roots of 𝑥 2 + 𝑥 − 6 = 0 satisfy the equation 𝑃(𝑥) = 0.
Now, 𝑥2 + 𝑥 − 6 = 0
⇨ 𝑥 2 + 3𝑥 − 2𝑥 − 6 = 0
⇨ 𝑥(𝑥 + 3) − 2(𝑥 + 3) = 0
⇨ (𝑥 + 3)(𝑥 − 2) = 0
⇨ 𝑥 = −3,2
∴ 𝑃(−3) = 0 ⇨ −27 + 9𝑃 + 𝑄 = 0………..(i)
and 𝑃(2) = 0 ⇨ 8 + 4𝑃 + 𝑄 = 0………..(ii)
Solving (i) and (ii):
we get 𝑃 = 7 and 𝑄 = −36
Nature of the Roots of a Quadratic Equation:
Let the quadratic equation be 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, 𝑎, 𝑏, 𝑐 ∈ ℝ and a≠ 0. The nature of the
roots of a quadratic equation is decided by the discriminant (i.e. 𝐷 = 𝑏 2 − 4𝑎𝑐)
(i) If 𝑏 2 − 4𝑎𝑐 > 0, then the quadratic equation has two real and distinct roots.
−𝑏
(ii) If 𝑏 2 − 4𝑎𝑐 = 0, then the quadratic equation has two equal roots i.e. 𝛼 = 𝛽 = .
2𝑎
(iii) If 𝑏 2 − 4𝑎𝑐 < 0, then the quadratic equation has two distinct complex roots, namely
−𝑏+𝑖√4𝑎𝑐−𝑏2 −𝑏−𝑖√4𝑎𝑐−𝑏2
𝛼= and 𝛽 =
2𝑎 2𝑎
(iv) If 𝑎, 𝑏, 𝑐 ∈ ℚ and 𝐷 is a perfect square, then equation has rational roots.
(v) The roots are of the form 𝑝 + √𝑞 (𝑝, 𝑞𝜖ℚ) iff 𝑎, 𝑏, 𝑐 are rational and 𝐷 is not a perfect
square.
Topic:- QUADRATIC EQUATIONS
Definition:- When we equate quadratic polynomial of the form (𝑎𝑥 2 + 𝑏𝑥 + 𝑐) equal to
zero we get a quadratic equation, where a, b and c are real numbers and a≠ 0.
Polynomial and Polynomial Equation:
As an expression of the form 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−2 +. . . +𝑎𝑛−1 𝑥 + 𝑎𝑛 where 𝑎0 , 𝑎1 , 𝑎2 … 𝑎𝑛 are
constant (𝑎 ≠ 0) and n is a positive integer, then it is called a polynomial in 𝑥 of degree 𝑛.
If 𝑓(𝑥) is a real or complex polynomial, then 𝑓(𝑥) = 0 is known as a polynomial equation.
Eg. If 𝑥 2 + 3𝑥 + 2 is a real polynomial, then 𝑥 2 + 3𝑥 + 2 = 0 is a polynomial equation.
Quadratic Equation:
If 𝑓(𝑥) is a polynomial of degree 2, then 𝑓(𝑥) = 0 is called a quadratic equation. The
general form of a quadratic equation is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏 and 𝑐 are real
numbers and 𝑎 ≠ 0. Here, 𝑥 is the variable and 𝑎, 𝑏, 𝑐 are the real coefficients.
Roots of a Quadratic Equation:
The values of the variable satisfying the given quadratic equations are called roots of that
equation. In other words, 𝑥 = 𝛼 is a root of the equation, 𝑓(𝑥) = 0, if 𝑓(𝛼) = 0.
The set of all roots of an equation, in a given domain is called the solution set of the
equation. The quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, 𝑐 ∈ ℝ and 𝑎 ≠ 0 has two
roots, namely
−𝑏+√𝐷 −𝑏−√𝐷
𝛼= and 𝛽 =
2𝑎 2𝑎
where, 𝐷 = 𝑏 2 − 4𝑎𝑐 is called the discriminant.
, Example 1. If 𝑥 2 + 𝑥 − 6 is a factor of polynomial 𝑥 3 + 𝑃𝑥 2 + 𝑄, then what are the values
of 𝑃 and 𝑄?
Soln: Let 𝑃(𝑥) = 𝑥 3 + 𝑃𝑥 2 + 𝑄
Since 𝑥 2 + 𝑥 − 6 is a factor of 𝑃(𝑥).
∴ Roots of 𝑥 2 + 𝑥 − 6 = 0 satisfy the equation 𝑃(𝑥) = 0.
Now, 𝑥2 + 𝑥 − 6 = 0
⇨ 𝑥 2 + 3𝑥 − 2𝑥 − 6 = 0
⇨ 𝑥(𝑥 + 3) − 2(𝑥 + 3) = 0
⇨ (𝑥 + 3)(𝑥 − 2) = 0
⇨ 𝑥 = −3,2
∴ 𝑃(−3) = 0 ⇨ −27 + 9𝑃 + 𝑄 = 0………..(i)
and 𝑃(2) = 0 ⇨ 8 + 4𝑃 + 𝑄 = 0………..(ii)
Solving (i) and (ii):
we get 𝑃 = 7 and 𝑄 = −36
Nature of the Roots of a Quadratic Equation:
Let the quadratic equation be 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, 𝑎, 𝑏, 𝑐 ∈ ℝ and a≠ 0. The nature of the
roots of a quadratic equation is decided by the discriminant (i.e. 𝐷 = 𝑏 2 − 4𝑎𝑐)
(i) If 𝑏 2 − 4𝑎𝑐 > 0, then the quadratic equation has two real and distinct roots.
−𝑏
(ii) If 𝑏 2 − 4𝑎𝑐 = 0, then the quadratic equation has two equal roots i.e. 𝛼 = 𝛽 = .
2𝑎
(iii) If 𝑏 2 − 4𝑎𝑐 < 0, then the quadratic equation has two distinct complex roots, namely
−𝑏+𝑖√4𝑎𝑐−𝑏2 −𝑏−𝑖√4𝑎𝑐−𝑏2
𝛼= and 𝛽 =
2𝑎 2𝑎
(iv) If 𝑎, 𝑏, 𝑐 ∈ ℚ and 𝐷 is a perfect square, then equation has rational roots.
(v) The roots are of the form 𝑝 + √𝑞 (𝑝, 𝑞𝜖ℚ) iff 𝑎, 𝑏, 𝑐 are rational and 𝐷 is not a perfect
square.
