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If a + b - c = 5 and a ^ 2 + b ^ 2 + c ^ 2 = 29 find the value of ab - bc - ca

User Emanuelez
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We can start by using the identity:

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

Expanding the left-hand side of the equation gives:

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca + 2ab + 2bc - 2ca

Simplifying, we get:

(a + b + c)^2 = a^2 + b^2 + c^2 + 4ab + 4bc

We can now substitute the given values of a^2 + b^2 + c^2 and a + b - c into this equation to solve for ab + bc:

(a + b + c)^2 = a^2 + b^2 + c^2 + 4ab + 4bc

(5 + c)^2 = 29 + 4(ab + bc)

25 + 10c + c^2 = 29 + 4(ab + bc)

c^2 + 10c - 4(ab + bc) + 4 = 0

We can now use the quadratic formula to solve for c:

c = (-10 ± sqrt(10^2 - 4*(-4)(a^2 + b^2 + 2ab - 29))) / (2(-4))

Simplifying this expression gives:

c = (-5 ± sqrt(13)) / 2

We can now use the given equation a + b - c = 5 to solve for a + b:

a + b = c + 5

Substituting the two possible values of c, we get:

a + b = (-5 + sqrt(13)) / 2 + 5 = (5 + sqrt(13)) / 2

or

a + b = (-5 - sqrt(13)) / 2 + 5 = (5 - sqrt(13)) / 2

We can now use the identity ab - bc - ca = (a + b - c)(b + c - a)(c + a - b) to calculate the desired value:

ab - bc - ca = ((5 + sqrt(13)) / 2)((5 - sqrt(13)) / 2)(-c) + ((5 - sqrt(13)) / 2)(-c)((5 + sqrt(13)) / 2) + (-c)((5 + sqrt(13)) / 2)((5 - sqrt(13)) / 2)

Simplifying this expression gives:

ab - bc - ca = -13/2

Therefore, the value of ab - bc - ca is -13/2.

User Florian Humblot
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