- The general form of a quadratic equation is ax^2 + bx + c = 0. There are various methods to solve quadratic equations like factorizing, completing the square, or using the quadratic formula. - The discriminant (D) determines the nature of the roots - if D>0 then roots are real and distinct, if D=0 then roots are real and equal, if D<0 then roots are imaginary. - Several properties are described regarding the relationship between the signs of coefficients a, b, c and the nature of the roots. - Example problems are given to illustrate solving quadratic equations and proving properties about roots.

1. QUADRATIC EQUATIONS

2. • The general form of a quadratic equation is
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 (a≠ 0)
Method of solving
a) Factorisation b)completing the square
c)formula etc
The roots of the above equation are
𝑥 =
−𝑏± 𝑏2−4𝑎𝑐
2𝑎

3. • If 𝛼, 𝛽 are the roots of the quadratic equation
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0,then
𝛼 + 𝛽 = −
𝑏
𝑎
𝛼𝛽 =
𝑐
𝑎

4. • If 𝛼, 𝛽 are the roots of a quadratic equation,
then the quadratic equation will be 𝑥2
−
(𝛼 + 𝛽)𝑥 + 𝛼𝛽 = 0

5. • Nature of the roots of the quadratic equation
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 depends upon 𝑏2
−
4𝑎𝑐,which is called the discriminant(D) of the
quadratic equation.
• If D>0,the roots will be real and distinct.
• If D=0,the roots will be real and equal.
• If D<0,the roots will be imaginary.

6. • If D>0 and a perfect square then roots will be
rational(provided a,b,c are rational).
• If D>0 and not a perfect square then roots will
be of the form 𝑝 ± 𝑞 (provided a,b,c are
rational)
• The imaginary roots and irrational roots
occurs in conjugates.(unless…)

7. • If a quadratic equation in x has more than two
roots, then it is an identity in x ,that is
a=b=c=0.
• If a,b,c are all positive then both the roots will
be negative.
• If a,c are of opposite sign, then the roots will
be of opposite signs.

8. • If a,c are of same sign ,then both the roots will
be positive.
• If b=0,then the roots will be equal in
magnitude and opposite in sign.
• For two distinct real roots, one will be more
than -
𝑏
2𝑎
and the other will be more than that.

9. • Q.If the roots of the equation ( 𝑎 − 𝑏)𝑥2
+ (𝑏 − 𝑐)𝑥 + (𝑐 −
𝑎) = 0 are equal, prove that b + c= 2a.
• Q.Prove that the roots of the equation ( 𝑥 − 𝑎)(𝑥 − 𝑏) +
(𝑥 − 𝑏)(𝑥 − 𝑐) + (𝑥 − 𝑐)(𝑥 − 𝑎) = 0 are real but they are
equal only when a=b=c.
• Q.Find the value of k for which the equation 𝑥2 + 𝑘𝑥 +
64 = 0 and 𝑥2
− 8𝑥 + k = 0 will have real roots.
• Q.Solve the equations
a) 9𝑥2
− 9(𝑎 + 𝑏)𝑥 + (2𝑎2
+ 5𝑎𝑏 + 2𝑏2
) = 0

10. • b)
1
𝑎+𝑏+𝑥
=
1
𝑎
+
1
𝑏
+
1
𝑥
• Q.A sailor can row a boat 8k.m. down stream
and return back to the starting point in 1 hour
40 minutes. If the speed of the stream is
2k.m/ hour, find the speed of the boat in still
water.

11. • Two water taps together can fill a tank in 9
3
8
hours. The tap of larger diameter takes 10
hours less than the smaller one to fill the tank
separately. Find the time in which each tap
can separately fill the tank

12. • If the equation( 𝑘2
− 5𝑘 + 6)𝑥2
+ (𝑘2
− 3𝑘 +
2)𝑥 + (𝑘2
− 4) = 0 is satisfied by more than
two values of x, then find k.
• If the roots of the equation
1
𝑥+𝑝
+
1
𝑥+𝑞
=
1
𝑟
are
equal in magnitude but opposite in sign then
show that product of the roots is equal to
−
𝑝2+𝑞2
2

13. • If the roots of the equation (x-a)(x-b)-k=0 are
c,d then show that the roots of the equation
(x-c)(x-d)+k=0 are a,b.
• If 𝛼 ≠ 𝛽 and 𝛼2
= 5𝛼 − 3, 𝛽2
= 5𝛽 − 3,find
the equation whose roots are
𝛼
𝛽
,
𝛽
𝛼
.
• If one root of the equation 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0
is square of the other, then prove that 𝑏3
+
𝑎2
𝑐 + 𝑎𝑐2
= 3𝑎𝑏𝑐

14. • Find the condition for which the equations
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0, 𝑎/
𝑥2
+ 𝑏/
𝑥 + 𝑐/
= 0 will
have a)one root in common b)both the roots
common.
• Solve
• a) 𝑥 + 1 + 𝑥 − 1 = 1
• b) 2𝑥 − 2 + 𝑥 − 3 = 2