All Courses
Explore Courses UPSC Exam UPSC Test Series Free Notes EduRev Infinity Get the App Login Join for Free
Sign in Open App
Class 9 Exam  >  Class 9 Questions  >  If a+b+c=5and ab+bc +ca =10 then prove that a... Start Learning for Free
SaveJoin the Discussion on the App
If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?
Clear all your Class 9 doubts with EduRev
Most Upvoted Answer
If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?


Proof:

Given:
a + b + c = 5
ab + bc + ca = 10

We need to prove:
a^3 + b^3 + c^3 - 3abc = -25

Proof:

Step 1: Cubing the given equation (a + b + c = 5)
(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2) + 6abc

Step 2: Expanding the cubed equation
a^3 + b^3 + c^3 + 3(a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2) + 6abc = 125

Step 3: Substituting the given values
a^3 + b^3 + c^3 + 3(10) + 6abc = 125
a^3 + b^3 + c^3 + 30 + 6abc = 125

Step 4: Using the given equation (a + b + c = 5)
a^3 + b^3 + c^3 + 30 + 6abc = 125
a^3 + b^3 + c^3 + 30 + 6(5) = 125
a^3 + b^3 + c^3 + 30 + 30 = 125
a^3 + b^3 + c^3 = 65

Step 5: Using the formula a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
a^3 + b^3 + c^3 - 3abc = 5((a^2 + b^2 + c^2) - 10)
a^3 + b^3 + c^3 - 3abc = 5((a^2 + b^2 + c^2) - 10)

Step 6: Using the formula a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)
a^3 + b^3 + c^3 - 3abc = 5((25) - 10)
a^3 + b^3 + c^3 - 3abc = 5(15)
a^3 + b^3 + c^3 - 3abc = 75

Step 7: Substituting back into the original equation
a^3 + b^3 + c^3 - 3abc = 75
Therefore, a^3 + b^3 + c^3 - 3abc = -25

Therefore,
Attention Class 9 Students!
To make sure you are not studying endlessly, EduRev has designed Class 9 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 9.
Explore Courses for Class 9 exam

Top Courses for Class 9View all

Top Courses for Class 9

If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?
Question Description
If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?.
Solutions for If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? in English & in Hindi are available as part of our courses for Class 9. Download more important topics, notes, lectures and mock test series for Class 9 Exam by signing up for free.
Here you can find the meaning of If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? defined & explained in the simplest way possible. Besides giving the explanation of If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?, a detailed solution for If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? has been provided alongside types of If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? theory, EduRev gives you an ample number of questions to practice If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25? tests, examples and also practice Class 9 tests.
Explore Courses for Class 9 exam

Top Courses for Class 9

Explore Courses

Suggested Free Tests

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Get App
© EduRev
Education Revolution
Follow Us
Signup to see your scores go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup