Page 1
(Quadratic equation is an equation in which the highest power of an
unknown variable is 2)
1. Quadratic Equation:
Any equation of the form p(x) = 0, where p(x) is a polynomial of
degree/power 2, is a quadratic equation.
Example:
,
,
, etc.
2. Standard Form of Quadratic Equation:
When we write the terms of p(x) in descending order of their degrees,
then we get the standard form of the equation i.e.
,
where a, b, c are real numbers, a ? 0.
3. Methods to find roots/solutions of a quadratic
equation:
a. Factorization method
b. Completing the square method
c. Discriminant method
Page 2
(Quadratic equation is an equation in which the highest power of an
unknown variable is 2)
1. Quadratic Equation:
Any equation of the form p(x) = 0, where p(x) is a polynomial of
degree/power 2, is a quadratic equation.
Example:
,
,
, etc.
2. Standard Form of Quadratic Equation:
When we write the terms of p(x) in descending order of their degrees,
then we get the standard form of the equation i.e.
,
where a, b, c are real numbers, a ? 0.
3. Methods to find roots/solutions of a quadratic
equation:
a. Factorization method
b. Completing the square method
c. Discriminant method
4. Solution of a Quadratic Equation by Factorization:
a. A real number a is said to be a root of the quadratic equation
, if
we can say that x = a is a
solution of the quadratic equation
Note:
b. If we factorize
, a ? 0, into a product of two linear
factors, then the roots of the quadratic equation
can be found by equating each factor to zero.
Example:
The roots of
are the values of x for which
(3x – 2)(2x + 1) = 0
(3x – 2) = 0 or (2x + 1) = 0
Zeroes of the quadratic
polynomial ?? ??
???? ??
Roots of the quadratic
equation ?? ??
???? ??
Page 3
(Quadratic equation is an equation in which the highest power of an
unknown variable is 2)
1. Quadratic Equation:
Any equation of the form p(x) = 0, where p(x) is a polynomial of
degree/power 2, is a quadratic equation.
Example:
,
,
, etc.
2. Standard Form of Quadratic Equation:
When we write the terms of p(x) in descending order of their degrees,
then we get the standard form of the equation i.e.
,
where a, b, c are real numbers, a ? 0.
3. Methods to find roots/solutions of a quadratic
equation:
a. Factorization method
b. Completing the square method
c. Discriminant method
4. Solution of a Quadratic Equation by Factorization:
a. A real number a is said to be a root of the quadratic equation
, if
we can say that x = a is a
solution of the quadratic equation
Note:
b. If we factorize
, a ? 0, into a product of two linear
factors, then the roots of the quadratic equation
can be found by equating each factor to zero.
Example:
The roots of
are the values of x for which
(3x – 2)(2x + 1) = 0
(3x – 2) = 0 or (2x + 1) = 0
Zeroes of the quadratic
polynomial ?? ??
???? ??
Roots of the quadratic
equation ?? ??
???? ??
Note:
For equations with coefficient of ??
other than 1, divide the whole equation by
the same number on both the sides to get 1 as the coefficient of ??
and then start
the process of completing the square.
5. Solution of a Quadratic Equation by Completing The
Square:
Page 4
(Quadratic equation is an equation in which the highest power of an
unknown variable is 2)
1. Quadratic Equation:
Any equation of the form p(x) = 0, where p(x) is a polynomial of
degree/power 2, is a quadratic equation.
Example:
,
,
, etc.
2. Standard Form of Quadratic Equation:
When we write the terms of p(x) in descending order of their degrees,
then we get the standard form of the equation i.e.
,
where a, b, c are real numbers, a ? 0.
3. Methods to find roots/solutions of a quadratic
equation:
a. Factorization method
b. Completing the square method
c. Discriminant method
4. Solution of a Quadratic Equation by Factorization:
a. A real number a is said to be a root of the quadratic equation
, if
we can say that x = a is a
solution of the quadratic equation
Note:
b. If we factorize
, a ? 0, into a product of two linear
factors, then the roots of the quadratic equation
can be found by equating each factor to zero.
Example:
The roots of
are the values of x for which
(3x – 2)(2x + 1) = 0
(3x – 2) = 0 or (2x + 1) = 0
Zeroes of the quadratic
polynomial ?? ??
???? ??
Roots of the quadratic
equation ?? ??
???? ??
Note:
For equations with coefficient of ??
other than 1, divide the whole equation by
the same number on both the sides to get 1 as the coefficient of ??
and then start
the process of completing the square.
5. Solution of a Quadratic Equation by Completing The
Square:
6. Discriminant:
A discriminant of a quadratic equation determines whether the
quadratic equation
has real roots or not.
Note: Check point no. 7: Nature of the roots
DISCRIMINANT =
7. Quadratic Formula:
The roots of a quadratic equation
are given by
v
Where, DISCRIMINANT (Discriminant =
)
8. Nature of the roots:
A quadratic equation
has
a. Two distinct real roots, if
b. Two equal roots, if
c. No real roots, if
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