Answer :
To find [tex]\( ab + bc + ca \)[/tex] given [tex]\( a + b + c = 13 \)[/tex] and [tex]\( a^2 + b^2 + c^2 = 69 \)[/tex], we can use the following identity:
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). \][/tex]
First, we substitute the given values into this identity:
[tex]\[ (13)^2 = 69 + 2(ab + bc + ca). \][/tex]
Next, we calculate [tex]\( (13)^2 \)[/tex]:
[tex]\[ 169 = 69 + 2(ab + bc + ca). \][/tex]
Now, isolate [tex]\( 2(ab + bc + ca) \)[/tex] by subtracting 69 from both sides:
[tex]\[ 169 - 69 = 2(ab + bc + ca), \][/tex]
[tex]\[ 100 = 2(ab + bc + ca). \][/tex]
To solve for [tex]\( ab + bc + ca \)[/tex], we divide both sides of the equation by 2:
[tex]\[ ab + bc + ca = \frac{100}{2}, \][/tex]
[tex]\[ ab + bc + ca = 50. \][/tex]
Thus, the value of [tex]\( ab + bc + ca \)[/tex] is 50.
The correct answer is:
b) 50
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). \][/tex]
First, we substitute the given values into this identity:
[tex]\[ (13)^2 = 69 + 2(ab + bc + ca). \][/tex]
Next, we calculate [tex]\( (13)^2 \)[/tex]:
[tex]\[ 169 = 69 + 2(ab + bc + ca). \][/tex]
Now, isolate [tex]\( 2(ab + bc + ca) \)[/tex] by subtracting 69 from both sides:
[tex]\[ 169 - 69 = 2(ab + bc + ca), \][/tex]
[tex]\[ 100 = 2(ab + bc + ca). \][/tex]
To solve for [tex]\( ab + bc + ca \)[/tex], we divide both sides of the equation by 2:
[tex]\[ ab + bc + ca = \frac{100}{2}, \][/tex]
[tex]\[ ab + bc + ca = 50. \][/tex]
Thus, the value of [tex]\( ab + bc + ca \)[/tex] is 50.
The correct answer is:
b) 50
