Question

find the value of ab + bc + ca if , a + b + c = 15 and
${a}^{2} + {b}^{2} + {c}^{2} is = 77$
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Answer 1

$\bf{\underline{\underline{\blue{Question:-}}}}$

• find the value of ab + bc + ca if , a + b + c = 15 and a² + b² + c² = 77

$\bf{\underline{\underline{\blue{Find:-}}}}$

• value of ab + bc + ca = ?

$\bf{\underline{\underline{\blue{Given:-}}}}$

• a + b + c = 15
• a² + b² + c² = 77

$\bf{\underline{\underline{\blue{Formula\: used:-}}}}$

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

$\bf{\underline{\underline{\blue{Calculation:-}}}}$

$\sf → (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$

$\sf → (15)^2=77+2(ab+bc+ca)$

$\sf → 225=77+2(ab+bc+ca)$

$\sf → 225-77=2(ab+bc+ca)$

$\sf → 148=2(ab+bc+ca)$

$\sf → ab+bc+ca=\dfrac{\cancel{148}}{\cancel{2}}$

$\sf → ab+bc+ca=74$

$\bf{\underline{\underline{\blue{Hence:-}}}}$

• value of ab + bc + ca = 74

Answer 2

Given,

$\green{ \boxed{ \rm{}a + b + c \gray = \purple{15}}}$

$\green{ \boxed{ \rm{a}^{2} + {b}^{2} + {c}^{2} \gray= \purple{77}}}$

$\bull \: \: \text{To \: find} \bf \: : \: \: \orange{ab + bc + ca}$

Here,

$\tt{}a + b + c \gray = 15 \\ \small\boxed{ \rm{squaring \: \: both \: \: sides}} \\ \to \tt{ {(a + b + c)}^{2} } = {(15)}^{2} \\ \small\to \tt{ {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ca) }= 225 \\ \small\boxed{ \rm{putting \: \: the \: \: known \: \: value}} \\ \to \tt{77 + 2(ab + bc + ca)} = 225 \\ \small\boxed{ \rm{transposing \: \: 77 \: \: to \: \: RHS}} \\ \ \to \tt2(ab + bc + ca) = 225 - 77 \\ \to \tt2(ab + bc + ca) = 148 \\ \small\boxed{ \rm{dividing \: \: both \: \: sides \: \: by \: \: 2 }} \\ \to \tt\frac{ \cancel2(ab + bc + ca)}{ \cancel2} = \frac{148}{2} \\ \\ \to \tt{ab + bc + ca} = \red{ 74}$

Thus,

• ab + bc + ca = 74