Quadratic Equation Notes
Quadratic Equations
Equations
Quadratic
MODULE - I
Algebra
2
Notes
QUADRATIC EQUATIONS
Recall that an algebraic equation of the second degree is written in general form as ax 2 + bx + c = 0, a ≠ 0
It is called a quadratic equation in x. The coefficient ‘a’ is the first or leading coefficient, ‘b’ is the second or middle coefficient and ‘c’ is the constant term (or third coefficient).
For example, 7x² + 2x + 5 = 0,
5
1
x² + x + 1 = 0,
2
2
1
= 0, 2 x² + 7x = 0, are all quadratic equations.
2
In this lesson we will discuss how to solve quadratic equations with real and complex coefficients and establish relation between roots and coefficients. We will also find cube roots of unity and use these in solving problems.
3x² − x = 0, x² +
OBJECTIVES
After studying this lesson, you will be able to:
• solve a quadratic equation with real coefficients by factorization and by using quadratic formula; • find relationship between roots and coefficients;
• form a quadratic equation when roots are given; and
• find cube roots of unity.
EXPECTED BACKGROUND KNOWLEDGE
• Real numbers
• Quadratic Equations with real coefficients.
MATHEMATICS
39
Quadratic Equations
MODULE - I
Algebra
2.1 ROOTS OF A QUADRATIC EQUATION
The value which when substituted for the variable in an equation, satisfies it, is called a root
(or solution) of the equation.
If be one of the roots of the quadratic equation
Notes
then,
ax 2 + bx + c = 0, a ≠ 0
... (i)
a 2 + b + c = 0
In other words, x − is a factor of the quadratic equation (i)
In particular, consider a quadratic equation x2 + x − 6 = 0
...(ii)
If we substitute x = 2 in (ii), we get
L.H.S = 22 + 2 – 6 = 0
∴
L.H.S = R.H.S.
Again put x = − 3 in (ii), we get
L.H.S. = ( − 3)2 –3 –6 = 0
∴
L.H.S = R.H.S.
Again put x = − 1 in (ii) ,we get
L.H.S = ( − 1)2 + ( − 1) – 6 = –6 ≠ 0 = R.H.S.
∴ x = 2 and x = − 3 are the only values of x which satisfy the
Equations
Quadratic
MODULE - I
Algebra
2
Notes
QUADRATIC EQUATIONS
Recall that an algebraic equation of the second degree is written in general form as ax 2 + bx + c = 0, a ≠ 0
It is called a quadratic equation in x. The coefficient ‘a’ is the first or leading coefficient, ‘b’ is the second or middle coefficient and ‘c’ is the constant term (or third coefficient).
For example, 7x² + 2x + 5 = 0,
5
1
x² + x + 1 = 0,
2
2
1
= 0, 2 x² + 7x = 0, are all quadratic equations.
2
In this lesson we will discuss how to solve quadratic equations with real and complex coefficients and establish relation between roots and coefficients. We will also find cube roots of unity and use these in solving problems.
3x² − x = 0, x² +
OBJECTIVES
After studying this lesson, you will be able to:
• solve a quadratic equation with real coefficients by factorization and by using quadratic formula; • find relationship between roots and coefficients;
• form a quadratic equation when roots are given; and
• find cube roots of unity.
EXPECTED BACKGROUND KNOWLEDGE
• Real numbers
• Quadratic Equations with real coefficients.
MATHEMATICS
39
Quadratic Equations
MODULE - I
Algebra
2.1 ROOTS OF A QUADRATIC EQUATION
The value which when substituted for the variable in an equation, satisfies it, is called a root
(or solution) of the equation.
If be one of the roots of the quadratic equation
Notes
then,
ax 2 + bx + c = 0, a ≠ 0
... (i)
a 2 + b + c = 0
In other words, x − is a factor of the quadratic equation (i)
In particular, consider a quadratic equation x2 + x − 6 = 0
...(ii)
If we substitute x = 2 in (ii), we get
L.H.S = 22 + 2 – 6 = 0
∴
L.H.S = R.H.S.
Again put x = − 3 in (ii), we get
L.H.S. = ( − 3)2 –3 –6 = 0
∴
L.H.S = R.H.S.
Again put x = − 1 in (ii) ,we get
L.H.S = ( − 1)2 + ( − 1) – 6 = –6 ≠ 0 = R.H.S.
∴ x = 2 and x = − 3 are the only values of x which satisfy the