An expression in the form of ax^{2} + bx + c, where a,b,c are real number and a ≠ 0, is called a quadratic polynomial.
For example- 3x^{2} + 5x + 7, √2x^{2} + 5x + 2 etc.
When a quadratic expression equated to zero is called quadratic equations. Hence an equation in the form of ax^{2} + bx + c = 0,
where a and b are coefficients of x^{2} and x, respectively and c is a constant.
For example- 3x2 + 5x + 7 = 0, √2x2 + 5x + 2 = 0 etc.
Example: Which of the following is not a quadratic equation?
(a) x^{2} + 2x + 2 ( 3 - x ) = 0
(b) x ( x + 1 ) + 1 = ( x - 2 ) ( x - 5 )
(c) ( 2x - 1 ) ( x - 3 ) = ( x + 5 ) ( x - 1 )
(d) x^{3} + 4x^{2} - x + 1 = ( x - 2 )^{3}
Solution:- (b)
x ( x + 1 ) + 1 = ( x - 2 ) ( x - 5 )
or, x^{2} + x + 1 = x ( x - 5 ) - 2 ( x - 5 )
or, x^{2} + x + 1 = x^{2} - 5x - 2x + 10
or, x^{2} + x + 1 = x^{2} - 7x + 10
or, x + 1 = -7x + 10
or, 8x - 9 = 0, which is not a quadratic equation.
Derivation of Quadratic Formula
we know that, ax^{2} + bx + c = 0 is the standard form of quadratic equation
Dividing on both sides by 'a'
Taking square root on both sides ,
Where, a = coefficient of x
^{2} b = coefficient of x
c = constant term
Discriminant (D)
For the quadratic equation ax^{2} + bx + c = 0,
D = b^{2} - 4ac
Where, D is the symbol of discriminant and a and b is the coefficients of x^{2} and x and c is a constant.
Example: If x^{2} - 3x + 1 = 0, find the value of D.
Solution:- a = 1 , b = -3 and c = 1
D = b^{2} - 4ac
= (-3)^{2} - 4 × 1 × 1
= 9 - 4
= 5
Roots of a quadratic Equation :-
1.If D > 0, then quadratic equation has two distinct real roots given by
Where, α and β are symbol of the roots of quadratic equation.
Example: Find the roots of the given equation, x^{2} + 8x + 4 = 0
Solution: given equation,
x^{2} + 8x + 4 = 0
Comparing with ax^{2} + bx + c = 0 , we get a = 1 , b = 8 and c = 4
then, D = b^{2} - 4ac
= 8^{2} - 4 × 1 × 4
= 64 - 1
∴ D = 63
2. If D = 0, then quadratic equation has two equal roots given by
Example: If ax^{2} + bx + c = 0 has equal roots, then find the value of c.
Solution: ax^{2} + bx + c = 0
D = 0
b^{2} - 4ac = 0
b^{2} = 4ac
3. If D < 0, then quadratic equation has no real roots.
Example: 3x^{2} + 7x + 5 = 0
Solution: 3x^{2} + 7x + 5 = 0
Comparing with ax^{2} + bx + c = 0, we get a = 3, b = 7 and c = 5
D = b^{2} - 4ac
= 7^{2} - ( 4 × 3 × 5 )
= 49 - 60
= -11
∴ No real roots.
Sum and Product of Roots :-
Sum of roots :-
Product of roots :-
Example: Find the sum and product of the roots in the equation x^{2} + 6x + 9 = 0
Solution: given equation x^{2} + 6x + 9 = 0
Here, a = 1, b = 6 and c = 9
Important points
1.A quadratic equation has two and only two roots.
2. A quadratic equation can not have more than two different roots.
3. If D is a perfect square then roots are rational otherwise irrational.
4. If a + √b is one root of a quadratic equation, then its conjugate a -√b must be the other root.
Formation of an Equation with given roots
If α and β are the roots of a quadratic equation ax^{2} + bx + c = 0, then the quadratic equation will be
Given equation,
ax
^{2} + bx + c = 0
Dividing on both sides by a, we get
From eq. (1) and (2), we get
x
^{2} - ( sum of the roots ) x + Product of roots = 0
x^{2} - ( α + β ) x + α . β = 0
Example: If α and β are the roots of the equation ax^{2} + bx + c = 0, then find the quadratic equation whose root are ( 2 + √3 ) and ( 2 - √3 ).
Solution: given roots are 2 + √3 and 2 - √3
Sum of roots = 2 + √3 + 2 - √3 = 4
Product of roots = ( 2 + √3 ) × ( 2 - √3 ) = (2)^{2} - ( √3 )^{2}
= 4 - 3 = 1
Required equation,
x^{2} - ( α + β ) x + α . β = 0
x^{2} - 4x + 1 = 0
Example: If α and β are the roots of the equation 5x^{2} - 2x + 8 = 0, then find the quadratic equation whose roots are 1/α and 1/β.
Solution: given equation,
5x^{2} - 2x + 8 = 0
a = 5 , b = -2 and c = 8
Required equation,
x
^{2} - ( α + β ) x + α . β = 0