Quadratic Equations : Quadratic Equations Lesson 5: CBSE Text book, 2008
S Chand & Co
Definitions : Definitions Any equation of the form p(x)=o ; where p(x) is a polynomial of degree 2 is a quadratic equation.
i.e. ax2+bx+c=0 is a standard form of quadratic equation, where a≠0 Important: please bear in mind Note:
The quadratic Equation
should have the
second degree polynomial See Example 5.2 on page 174
What is a quadratic equation : What is a quadratic equation The power of the variable x should be = 2 ( not less or more)
The powers of the other variables should be less than 2 : but only as integers ax2+bx+c=0 is a standard form of quadratic equation, where a≠0 If k is the root of the quadratic eqn:
Then x-k is a factor of the equation:
Also substituting k in place of x shall be able to satisfy the condition of the equation
Some examples of Quadratic Equations : Some examples of Quadratic Equations
Understanding the Roots…(1) : Understanding the Roots…(1)
Understanding the Roots…(2) : Understanding the Roots…(2) We are comparing the coefficients
E5.1/2(v) : E5.1/2(v) Now check the other condition given in the problem
The hypotenuse of a right angled triangle is 6 mtrs more than twice the shortest side. The third side is two meters less than the hypotenuse : The hypotenuse of a right angled triangle is 6 mtrs more than twice the shortest side. The third side is two meters less than the hypotenuse h s l h = 2s+6 ----(1)
l = h-2 -----(2)
E5.3/21 : E5.3/21 Understand these steps:
We have to the split the middle term (5a) to enable the common elements for factorization Taking the common factor:a-2
E5.3/30 : E5.3/30 Two roots
Please remember the following : Please remember the following
Find the two numbers whose sum is 27 and the product 182 : Find the two numbers whose sum is 27 and the product 182 13 & 14 are known
as the roots of the
Quadratic Equation
x2-27x+182=0
Roots of Quadratic Equation: Proof : Roots of Quadratic Equation: Proof Only if b2≥4ac then only the quadratic equation has real roots
E5.4/9 : E5.4/9 Similarly x=0 can be shown
E5.4/20 : E5.4/20
E5.5/17 : E5.5/17
Summary : Summary
E5.7/14 : E5.7/14
E5.8/3 : E5.8/3
E5.8/5 : E5.8/5
Problem : Problem Two Water Taps together fill a tank in 9 3/8 hrs. The tap of the large diameter takes 10hrs less that the small diameter one to fill the tank separately. Find the time in which each tap can separately fill the tank
Solution --------(page1) : Solution --------(page1)
----(page2) : ----(page2)
E5.13/14 : E5.13/14
Find the value of k in the following eqn : Find the value of k in the following eqn
E5.11/5 : E5.11/5
E5.15/4 : E5.15/4
E5.16/8 : E5.16/8
Slide 29 : If α&β are the roots of the eqn ax2+bx+c=0 then find the value of
Slide 30 :
Slide 31 : Solve these yourself
Slide 32 :
Slide 33 : TIP: Expand p3+q3 as (p+q)(p2……..q2) Solve these yourself
Slide 34 : Check how!!!!
Slide 35 : Tips for solving the problems
Thanks for your attention : Thanks for your attention