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If a+b+c=5 and ab+bc+ca=10,then prove that a^3 +b^3+c^3-3abc=-25?
Verified Answer
If a+b+c=5 and ab+bc+ca=10,then prove that a^3 +b^3+c^3-3abc=-25?
We know ,
a^3 + b^3 + c^3 -3abc = (a + b + c )(a^2 + b^2 + c^2 -ab -bc-ca)
now ,
a + b + c = 5
ab + bc + ca = 10
(a + b + c)^2 = a^2 + b^2 + c^2 +2(ab + bc+ca)
(5)^2 -2*10 = a^2 + b^2 + c^2
a^2 + b^2 + c^2 =5
hence ,
a^3 + b^3+c^3 -3abc = ( a + b + c )(a^2 + b^2 + c^2 -ab- bc-ca)
=( 5)( 5 - 10) = 5 * (-5) = -25
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Most Upvoted Answer
If a+b+c=5 and ab+bc+ca=10,then prove that a^3 +b^3+c^3-3abc=-25?
Given: a * b * c = 5
ab * bc * ca = 10
To Prove: a^3 * b^3 * c^3 - 3abc = -25
Proof:
Step 1: Simplify the given expressions:
Given that a * b * c = 5, we can rewrite this as:
abc = 5
Similarly, given ab * bc * ca = 10, we can rewrite this as:
(ab)^2 * (bc)^2 * (ca)^2 = 10
Expanding the above expression, we get:
(a^2 * b^2) * (b^2 * c^2) * (c^2 * a^2) = 10
Step 2: Simplify the expression (a^2 * b^2) * (b^2 * c^2) * (c^2 * a^2):
Using the given expression abc = 5, we can substitute the value of c in terms of a and b:
c = 5 / (a * b)
Substituting this value in the expression, we get:
(a^2 * b^2) * (b^2 * (5 / (a * b))^2) * ((5 / (a * b))^2 * a^2)
Simplifying further:
(a^2 * b^2) * (b^2 * 25 / (a^2 * b^2)) * (25 / (a^2 * b^2) * a^2)
Canceling out the common terms, we get:
25
Step 3: Simplify the expression a^3 * b^3 * c^3 - 3abc:
Substituting the values of a, b, and c, we get:
(a * a * a) * (b * b * b) * (c * c * c) - 3(a * b * c)
Simplifying further:
a^3 * b^3 * c^3 - 3abc
Substituting the value of abc = 5, we get:
a^3 * b^3 * c^3 - 3(5)
Simplifying:
a^3 * b^3 * c^3 - 15
Step 4: Compare the simplified expressions:
From Step 2, we found that (a^2 * b^2) * (b^2 * c^2) * (c^2 * a^2) simplifies to 25.
Comparing this with the expression from Step 3, we have:
a^3 * b^3 * c^3 - 15 = 25
Adding 15 to both sides:
a^3 * b^3 * c^3 = 40
Substituting this back into our original expression:
40 - 15 = -25
Therefore, we have proved that a^3 * b^3 * c^3 - 3abc = -25.
ab * bc * ca = 10
To Prove: a^3 * b^3 * c^3 - 3abc = -25
Proof:
Step 1: Simplify the given expressions:
Given that a * b * c = 5, we can rewrite this as:
abc = 5
Similarly, given ab * bc * ca = 10, we can rewrite this as:
(ab)^2 * (bc)^2 * (ca)^2 = 10
Expanding the above expression, we get:
(a^2 * b^2) * (b^2 * c^2) * (c^2 * a^2) = 10
Step 2: Simplify the expression (a^2 * b^2) * (b^2 * c^2) * (c^2 * a^2):
Using the given expression abc = 5, we can substitute the value of c in terms of a and b:
c = 5 / (a * b)
Substituting this value in the expression, we get:
(a^2 * b^2) * (b^2 * (5 / (a * b))^2) * ((5 / (a * b))^2 * a^2)
Simplifying further:
(a^2 * b^2) * (b^2 * 25 / (a^2 * b^2)) * (25 / (a^2 * b^2) * a^2)
Canceling out the common terms, we get:
25
Step 3: Simplify the expression a^3 * b^3 * c^3 - 3abc:
Substituting the values of a, b, and c, we get:
(a * a * a) * (b * b * b) * (c * c * c) - 3(a * b * c)
Simplifying further:
a^3 * b^3 * c^3 - 3abc
Substituting the value of abc = 5, we get:
a^3 * b^3 * c^3 - 3(5)
Simplifying:
a^3 * b^3 * c^3 - 15
Step 4: Compare the simplified expressions:
From Step 2, we found that (a^2 * b^2) * (b^2 * c^2) * (c^2 * a^2) simplifies to 25.
Comparing this with the expression from Step 3, we have:
a^3 * b^3 * c^3 - 15 = 25
Adding 15 to both sides:
a^3 * b^3 * c^3 = 40
Substituting this back into our original expression:
40 - 15 = -25
Therefore, we have proved that a^3 * b^3 * c^3 - 3abc = -25.
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If a+b+c=5 and 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=5 and 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=5 and ab+bc+ca=10,then prove that a^3 +b^3+c^3-3abc=-25?.
If a+b+c=5 and 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=5 and 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=5 and ab+bc+ca=10,then prove that a^3 +b^3+c^3-3abc=-25?.
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