Question

if a square
$if \: a ^{2} + b ^{2} + c ^{2} = 25 \: and \: ab + bc + ca = 3 \: h \\ then \: a + b + c = ?$
if a square + b square + c square =25 and ab+bc+ca=3 then a+b+c=?​

Answer 1

Algebraic Identities

The following algebraic identity will be used to find the solution:

$\boxed{(a+b+c)^2 = a^2 + b^2 + c^2 +2(ab + bc + ca)}$

We have been given that, $a ^{2} + b ^{2} + c ^{2} = 25$ and $ab + bc + ca = 3$. With this information, we have been asked to find out the value of $a + b + c$.

We know that,

$\boxed{(a+b+c)^2 = a^2 + b^2 + c^2 +2(ab + bc + ca)}$

By substituting the known values in the above identity, we get:

$\implies (a+b+c)^2 = 25 + 2(3) \\ \\ \implies (a+b+c)^2 = 25 + 6 \\ \\ \implies (a+b+c)^2 = 31 \\ \\ \implies a+b+c = \sqrt{31} \\ \\ \implies \boxed{a+b+c = 5.56}$

Hence, the value a + b + c is 5.56.

$\rule{90mm}{2pt}$

MORE TO KNOW

$\boxed{\begin{array}{l}\boxed{\bigstar\:\:\textbf{\textsf{Algebric\:Identity}}\:\bigstar}\\\\\frak{1.}\bf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\\frak{2.}\sf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\\frak{3.}\bf\: A^{2} - B^{2} = (A+B)(A-B)\\\\\frak{4.}\sf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\\frak{5.}\bf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\\frak{6.}\sf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\\frak{7.}\bf\:(A-B)^{3} = A^{3} - 3AB(A-B) + B^{3}\\\\\frak{8.}\sf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2})\\\\\end{array}}$

Answer 2

Answer:

Given :-

• a² + b² + c² = 25
• ab + bc + ca = 3

To Find :-

• What is the value of a + b + c ?

Solution :-

Given :

$\leadsto \bf a^2 + b^2 + c^2 =\: 25$

$\leadsto \bf ab + bc + ca =\: 3$

As we know that :

$\footnotesize \bigstar\: \: \sf\boxed{\bold{\pink{(a + b + c)^2 =\: a^2 + b^2 + c^2 + 2(ab + bc + ca)}}}\: \: \bigstar$

So, according to the question by using the formula we get,

$\footnotesize \dashrightarrow \bf (a + b + c)^2 =\: a^2 + b^2 + c^2 + 2(ab + bc + ca)$

By putting :

• a² + b² + c² = 25
• ab + bc + ca = 3

$\dashrightarrow \sf (a + b + c)^2 =\: 25 + 2(3)$

$\dashrightarrow \sf (a + b + c)^2 =\: 25 + 2 \times 3$

$\dashrightarrow \sf (a + b + c)^2 =\: 25 + 6$

$\dashrightarrow \sf (a + b + c)^2 =\: 31$

$\dashrightarrow \sf a + b + c =\: \sqrt{31}$

$\dashrightarrow \sf a + b + c =\: 5.567$

$\dashrightarrow \sf\bold{\red{a + b + c =\: 5.567}}$

$\therefore$ The value of a + b + c is 5.567 .

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IMPORTANT ALGEBRAIC EXPRESSION :-

$\clubsuit \: \: \sf\bold{\pink{(a + b)^2 =\: a^2 + 2ab + b^2}}$

$\clubsuit \: \: \sf\bold{\pink{(a - b)^2 =\: a^2 - 2ab + b^2}}$

$\clubsuit \: \: \sf\bold{\pink{a^2 - b^2 =\: (a + b)(a - b)}}$

$\clubsuit \: \: \sf\bold{\pink{(x + a)(x + b) =\: x^2 + (a + b)x + ab}}$

$\clubsuit \: \: \sf\bold{\pink{(a + b + c)^2 =\: a^2 + b^2 + c^2 + 2ab + 2bc + 2ca}}$

$\clubsuit \: \: \sf\bold{\purple{(a + b)^3 =\: a^3 + 3a^2b + 3ab^2 + b^3}}$

$\clubsuit \: \: \sf\bold{\purple{(a - b)^3 =\: a^3 - 3a^2b + 3ab^2 - b^3}}$

$\clubsuit \: \: \sf\bold{\purple{a^3 + b^3 =\: (a + b)(a^2 + ab + b^2)}}$

$\footnotesize \clubsuit \: \: \sf\bold{\purple{a^3 + b^3 + c^3 - 3abc =\: (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)}}$