Understanding the Algebra Problem

We are given two equations involving three variables, \(a\), \(b\), and \(c\):

1. \(a + b + c = 6\)
2. \(a^2 + b^2 + c^2 = 14\)

Our goal is to find the value of the expression \((a - b)^2 + (b - c)^2 + (c - a)^2\). This problem requires us to use algebraic identities to relate the given information to the expression we need to evaluate.

Expanding the Expression

Let's expand each term in the expression \((a - b)^2 + (b - c)^2 + (c - a)^2\) using the identity \((x - y)^2 = x^2 - 2xy + y^2\):

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

Now, let's add these expanded terms together:

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

Combine like terms:

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

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

Factor out 2:

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

We can see that the expression depends on the values of \((a^2 + b^2 + c^2)\) and \((ab + bc + ca)\). We are given the value of \((a^2 + b^2 + c^2)\), which is 14. We need to find the value of \((ab + bc + ca)\).

Finding the Value of ab + bc + ca

We can find the value of \((ab + bc + ca)\) using the identity for the square of a sum of three terms: \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\).

We are given \(a + b + c = 6\) and \(a^2 + b^2 + c^2 = 14\). Substitute these values into the identity:

\((6)^2 = 14 + 2(ab + bc + ca)\)

Calculate the square of 6:

\(36 = 14 + 2(ab + bc + ca)\)

Subtract 14 from both sides:

\(36 - 14 = 2(ab + bc + ca)\)

\(22 = 2(ab + bc + ca)\)

Divide by 2 to find the value of \((ab + bc + ca)\):

\(ab + bc + ca = \frac{22}{2}\)

\(ab + bc + ca = 11\)

Calculating the Final Expression Value

Now that we have the values for \((a^2 + b^2 + c^2)\) and \((ab + bc + ca)\), we can substitute them back into the expanded expression for \((a - b)^2 + (b - c)^2 + (c - a)^2\):

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

Substitute \(a^2 + b^2 + c^2 = 14\) and \(ab + bc + ca = 11\):

\(= 2(14) - 2(11)\)

Perform the multiplication:

\(= 28 - 22\)

Perform the subtraction:

\(= 6\)

So, the value of \((a - b)^2 + (b - c)^2 + (c - a)^2\) is 6.

Summary of Steps

1. Expand the target expression \((a - b)^2 + (b - c)^2 + (c - a)^2\) using the identity \((x-y)^2 = x^2 - 2xy + y^2\).
2. Simplify the expanded expression to get it in terms of \((a^2 + b^2 + c^2)\) and \((ab + bc + ca)\).
3. Use the identity \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\) and the given values to find \((ab + bc + ca)\).
4. Substitute the calculated value of \((ab + bc + ca)\) and the given value of \((a^2 + b^2 + c^2)\) into the simplified expanded expression from step 2.
5. Calculate the final numerical value.

Revision Table: Key Algebraic Identities

Additional Information: Applying Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. They are powerful tools for simplifying expressions and solving equations.

• The identity \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\) is particularly useful when you are given the sum of variables and the sum of their squares, and you need to find the sum of their pairwise products (like \(ab + bc + ca\)).
• The identity \((a - b)^2 = a^2 - 2ab + b^2\) is fundamental for expanding squared differences. When you have a sum of squared differences like \((a - b)^2 + (b - c)^2 + (c - a)^2\), expanding and simplifying often reveals a structure related to sums of squares and sums of pairwise products.
• Problems like this demonstrate how different algebraic identities can be combined to solve a problem. By expanding the target expression and comparing it to the given information using another identity, we can bridge the gap and find the required value.

Mastering these identities is crucial for success in algebra.