**Blog-65**

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Dear Students,

Download following files for learning Formulas more effectively. Take the print out of these files and write your answers daily to improve your scores in 10th standard/grade.

[ Note: The following 2 files are available in the secured drive. While downloading, your email might be asked. Please provide it to download the files. I assure that your email-id will not be given to anybody ]

Following two files are very important to improve your base for remembering the formulas. It is 100 % free for all the students of the world. Download these two files, take the Print Out of these files on two sides of the paper and practice these formulas for your benefit to get high scores in Mathematics Standard 10/ Grade 10.

Please send the request for downloading the following files which will really help you to improve your mathematical skills for Standard 10 / Grade 10. This is the PRACTICE SHEET of Formulas.

In the Previous Blog we had seen some important concepts of Quadratic Equation.

1) Click Here to Read the Blog on (01 Basics of Quadratic Equations (Grades 9 to 12) Part-01)

4) Click Here to Read the Blog on (04 Basics of Quadratic Equations (Grades 9 to 12) Part-04)

5) Click Here to Read the Blog on (05 Basics of Quadratic Equations (Grades 9 to 12) Part-05)

5) Click Here to Read the Blog on (05 Basics of Quadratic Equations (Grades 9 to 12) Part-05)

Some special and critical types of the factors:

Please download the following file and study it very carefully so that you will not find any difficulties while solving quadratic equations.

Click Here to download the file " Critical-type-of-factors.pdf "

Just go through this downloaded file and be prepared to solve any problem pertaining to these critical factors.

Click Here to download the file " Critical-type-of-factors.pdf "

Just go through this downloaded file and be prepared to solve any problem pertaining to these critical factors.

Now we will see few quadratic equation using formula method.

a) Solve the quadratic equation 5 x

^{2}- 2 x - 9 = 0.
Solution:

1) Comparing 5 x

^{2}- 2 x - 9 = 0 with a x^{2}+ b x + c = 0 we get,
a = 5, b = - 2, c = - 9

b) Solve the quadratic equation 3 x

^{2}- 4 x - 3 = 0.
Solution:

1) Comparing 3 x

^{2}- 4 x - 3 = 0 with a x^{2}+ b x + c = 0 we get,
a = 3, b = - 4, c = - 3

Now we will see some rules about the roots of the quadratic equation.

1) Sum of the roots (α, β) of the quadratic equation a x

^{2}+ b x + c = 0,
α + β = - b / a and α β = c / a

α + β = - Coefficient of x / Coefficient of x

^{2}and α β = Constant / Coefficient of x^{2.}
2) If α, β are the roots of the quadratic equation then the quadratic equation will be

x

^{2}- (Sum of the roots) x + (Product of the roots) = 0,
x

^{2}- (α + β) x + (α β) = 0,^{Now we will see some important problems on Quadratic equation and the roots of the Quadratic Equations}

^{ }

^{ a) If the sum of the roots of the quadratic equation is 3 and the sum of their cubes is 63, then find the equation.}

^{ Solution:}

1) Here,

^{ }α + β = 3 and α^{3}+ β^{3}= 63
2) We know that ,

^{ }α^{3}+ β^{3}= (α + β)^{3 }- 3 α β (α + β)
so ,

^{ }63 = (3)^{3 }- 3 α β (3)
63 = 27

^{ }- 9 α β Dividing by 9 through out we get
7 = 3

^{ }- α β
α β = 3

^{ }- 7
α β = - 4

3) We know that the quadratic equation will be

x

^{2}- (α + β) x + (α β) = 0
x

^{2}- (3) x + (- 4) = 0
x

^{2}- 3 x - 4 = 0
4) Answer: So the required quadratic equation will be x

^{2}- 3 x - 4 = 0.
Few more problems will be discussed in the next Blog.