This document discusses quadratic equations. It begins by defining a quadratic equation as an equation with one variable where the highest power of that variable is 2. Some examples of quadratic equations are provided. It then discusses different methods for solving quadratic equations, including factorizing, using the quadratic formula, and word problems involving quadratic equations. Key steps in the methods are outlined such as expressing the equation in standard form and setting each factor equal to 0 when factorizing. Practice problems are provided to illustrate the different solving techniques.

1. X
MATHEMATICS
GOVERNMENT OF KERALA
DEPARTMENT OF EDUCATION

2. MATHEMATICS
Standard x
Government of Kerala
Department of education

3. Dear children,
In farms and factories,
Up the sky and in the mind.
Mathematics blooms.
Roots deep in history;
Numbers, equations,
Geometrical shapes,
Forking braches.
To know a little of all this,
A small book.
Fruit of knowledge- a mellow mind
Right in thought, True in word,
With regards,
Director SCERT

4. 1. QUADRATIC
EQUATION
INTRODUCTION
An equation, which contains one unknown (variable) with
highest power 2, is called a quadratic equation.
In other words, a quadratic equation is a second degree equation in
one variable.
e.g. (1) X^2 - 5x + 7 = 0
(2) 5x^2- 7x = 0
(3) 3x^2 – 8 = 0 and so on.
The standard form of a quadratic equation is ax^2 + bx + c = 0; where a, b and c
are all real numbers and a = 0.

5. QUADRATIC EQUATION
What is a quadratic equation
A quadratic equation is one variable is an equation in which the highest
power of the variable is two.
Thus 3x^2 + 2x – 1 = 0 is a quadratic equation in x. The standard form of a
quadratic equation in one variable is
Ax^2 + bx + c =0; a, b, c E R, a = 0
Thus, 5x^2 – 7x +8 = 0 is in standard form
Here, a = 5, b =- 7, c=8.
Solving quadratic equation by factorizing.
We follow the following two rules.
RULE 1: Every quadratic equation has two roots.
Thus, x^2 = 16 has two roots, 4 and -4,
That is x = +_ 4
In the case of the quadratic equation x^2 = 0,

6. The equation is considered two have two equal roots, each equal to 0,
while the quadratic equation (x – 4) = 0 is considered to have two equal
roots, each equal to 4.
RULE 2: If the product of two factors is zero then one or the other of
the factors equals zero.
Thus, is x (3x – 5) = 0, then x = 0 or 3x – 5 =0
Also, if (x – 2) (4x + 9) = 0, then x – 2 = 0 or 4x + 9 = 0.
To solve a quadratic equation, we work as under:
STEP 1: Express the given equation in the form ax^2 + bx + c = 0
STEP 2: Factorises ax^2 + bx + c
STEP 3: Put each factor = 0
STEP 4: Solve resulting equation
EXAMPLE:
Solve: 1
X^2-3x=0 x(x-3) =0
X=0 or x-3=0 x=0 or x=3
Solve: 2
X^2 +2x-15=0 x^2+5x-3x-15=0

7. X(x+5)-3(x+5) =0 (x+5) (x-3) =0
x+5=0 or x-3=0 x= (-5) x=3
EXERCISE
Solve
 (x-3)(x-7)=0
 (3x+4)(2x-11)=0
SOLVING QUADRATIC EQUATION BY USING FORMULA
(1) Let the given quadratic equation by using formula be ax^2+bx+c=0,a not
equal to 0.
SOLUTION:
a x^2+bx+c=0 (‘a’ not equal to 0)
Or ax^2+bx=-c (transposing the constant term)
Or x^2+ (b/a)x=-c/a (dividing by the coefficient of x^2)
(Adding b^2/4a^2 both sides to make L.H.S perfect square)
X^2+(b/a)x+b^2/4a^2= b^2/4a^2-c/a
Or(x +b/2a) ^2=b^2-4ac/4a^2 x+ b/2a=+or- root of b^2-4ac/2a
(Taking square root of both sides)
X=-b/2a +or- root of b^2-4ac/2a
X=-b+ or-root ofb^2-4ac/2a

8. EXERSICE
 3x^2 -10x+3=0
 6y^2-35y+50=0
 2x+4/x=9
WORD PROBLEMS INVOLVING QUADRATIC EQUATINS
EXAMPLE:
Find to consecutive positive even integers whose product is 224.
Solution: Let the two consecutive positive even integers be 2x and 2x+2.
Then, (2x) (2x+2)=224 4x^2+4x=224 x^2+x-56=0
(X-7)(x+8)=0 x=7 or x= (-8)
When x=7, we obtain the two consecutive positive integers as 14 and 16, where
as if x=(-8),we get (-16)and (-14) which are negative integers and are not
consistent with our requirement. There for, the integers are 14 and 16.
EXERCISE:
 The sum of two numbers is 15. If the sum of their reciprocals is 3/10, find
the two numbers.
 A number consist of two digits whose product is 18. When 27 is subtracted
from the number, the digit change their places. Find the number