We are given that \(a, b, c\) are non-zero numbers such that \(a + b + c = 0\). We need to find the Value of expression \(\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}\).

Finding the Value of the Algebraic Expression

The given expression is:

\[ \frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} \]

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of the denominators \(bc\), \(ac\), and \(ab\) is \(abc\). We can rewrite each term with the denominator \(abc\):

• The first term \(\frac{a^2}{bc}\) needs to be multiplied by \(\frac{a}{a}\). So, \(\frac{a^2}{bc} = \frac{a^2 \times a}{bc \times a} = \frac{a^3}{abc}\).
• The second term \(\frac{b^2}{ac}\) needs to be multiplied by \(\frac{b}{b}\). So, \(\frac{b^2}{ac} = \frac{b^2 \times b}{ac \times b} = \frac{b^3}{abc}\).
• The third term \(\frac{c^2}{ab}\) needs to be multiplied by \(\frac{c}{c}\). So, \(\frac{c^2}{ab} = \frac{c^2 \times c}{ab \times c} = \frac{c^3}{abc}\).

Now, we can add the modified terms:

\[ \frac{a^3}{abc} + \frac{b^3}{abc} + \frac{c^3}{abc} = \frac{a^3 + b^3 + c^3}{abc} \]

Using the Condition \(a + b + c = 0\)

We are given the condition \(a + b + c = 0\). There is a useful algebraic identity related to the sum of cubes when the sum of the numbers is zero. The identity states that if \(a + b + c = 0\), then \(a^3 + b^3 + c^3 = 3abc\).

We can verify this identity:

From \(a + b + c = 0\), we have \(a + b = -c\).

Cubing both sides:

\[ (a + b)^3 = (-c)^3 \]

\[ a^3 + b^3 + 3ab(a + b) = -c^3 \]

Substitute \((a + b) = -c\):

\[ a^3 + b^3 + 3ab(-c) = -c^3 \]

\[ a^3 + b^3 - 3abc = -c^3 \]

Rearranging the terms:

\[ a^3 + b^3 + c^3 = 3abc \]

This confirms the identity.

Simplifying the Expression to Find the Value

Now we can substitute the identity \(a^3 + b^3 + c^3 = 3abc\) into our expression \(\frac{a^3 + b^3 + c^3}{abc}\).

\[ \frac{a^3 + b^3 + c^3}{abc} = \frac{3abc}{abc} \]

Since \(a, b, c\) are non-zero, their product \(abc\) is also non-zero. Therefore, we can cancel \(abc\) from the numerator and the denominator to simplify the expression.

\[ \frac{3abc}{abc} = 3 \]

Thus, the Value of expression \(\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}\) is 3.

This problem demonstrates how a given equation (\(a+b+c=0\)) can be used with algebraic identities to find the Value of expression. Remember that the condition that \(a, b, c\) are non-zero numbers is important because it ensures that the denominators \(bc, ac, ab, abc\) are not zero, allowing division.

Conclusion

Given \(a, b, c\) are non-zero numbers and \(a + b + c = 0\), the Value of expression \(\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}\) simplifies to 3.