Question

If
${a }^{2}+{b}^{2} + {c}^{2} = 250$
and
$ab + bc + ca = 3$
then find
$a + b + c = x$
find the value of x​

Answer 1

GIVEN :–

$\\ \tt \implies{a}^{2}+{b}^{2} + {c}^{2} = 250$

$\\ \tt \implies ab + bc + ca = 3$

$\\ \tt \implies a+b+c=x$

TO FIND :–

• Value of 'x' = ?

SOLUTION :–

$\\ \tt \implies a+b+c=x$

• Now square on both sides –

$\\ \tt \implies (a+b+c)^{2} = {(x)}^{2}$

• We know that –

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

• So that –

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

• Now put the values –

$\\ \tt\implies 250+ 2(3) = {x}^{2}$

$\\ \tt\implies {x}^{2} = 250+ 2(3)$

$\\ \tt\implies {x}^{2} = 250+6$

$\\ \tt\implies {x}^{2} = 256$

$\\ \tt\implies x= \sqrt{256}$

$\\ \large\implies{ \boxed{ \tt x= \pm16}}$

Answer 2

Here, as per the provided question we have to find the value of x when the provided question which information is given there –

GIVEN :

The provided value (expression) are given here as follows :

• a² + b² + c² = 250 (gained value)

• ab + bc + ca = 3 (gained value)

• a + b + c = x (gained value)

TO FIND :

Now, in the third case given as a + b + c = x

• So, in the third case we have to find the value of x = ?

STEP -BY-STEP EXPLAINATION :

Here, now we will apply an appropriate formula for finding this [ squaring on both side ] –

$⟹ \rm ({a \: + \: b \: + c})^{2} \: = ( {x})^{2}$

Formula applied for solving the sum –

• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Now, after after substituting this formula into simpler form by removing bracket –

[ we will directly start solving the sum as we know how to substitute it ] –

⟹ 250 + 2(3) = x²

⟹ x² = 250 + 2(3)

⟹ x² = (250 + 6)

⟹ x² = 256

$⟹ \rm {x} \: = \sqrt{256}$

⟹ x = ±16

Therefore, the value of x we have obtained after solving this = ±16.