−44*A*2+12*AB*+8*B*2−7*AC*+*BC*−*C*2=0 solve by c

To solve the given equation for \( C \), we'll first rewrite it in a more readable form:

\[
-44A^2 + 12AB + 8B^2 - 7AC + BC - C^2 = 0
\]

This is a quadratic equation in terms of \( C \). Let's reorganize it as:

\[
-C^2 + (B - 7A)C + (-44A^2 + 12AB + 8B^2) = 0
\]

Now, this equation can be solved using the quadratic formula:

\[
C = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where:
- \( a = -1 \) (coefficient of \( C^2 \))
- \( b = B - 7A \) (coefficient of \( C \))
- \( c = -44A^2 + 12AB + 8B^2 \) (constant term)

Substitute these values into the formula:

\[
C = \frac{-(B - 7A) \pm \sqrt{(B - 7A)^2 - 4(-1)(-44A^2 + 12AB + 8B^2)}}{2(-1)}
\]

Simplify further:

\[
C = \frac{-(B - 7A) \pm \sqrt{(B - 7A)^2 - 4(-44A^2 + 12AB + 8B^2)}}{-2}
\]

We can simplify the expression inside the square root and calculate the discriminant.

Let's compute the values step-by-step.The two possible solutions for \( C \) are:

\[
C_1 = \frac{-7A + B - \sqrt{-127A^2 + 34AB + 33B^2}}{2}
\]

\[
C_2 = \frac{-7A + B + \sqrt{-127A^2 + 34AB + 33B^2}}{2}
\]

These represent the two possible values of \( C \) in terms of \( A \) and \( B \).

Would you like a deeper explanation of how these solutions were derived or have any other questions?

### Related Questions:
1. How do we determine if the solutions for \( C \) are real or complex?
2. What does the discriminant tell us about the nature of the roots?
3. Can we solve for \( C \) if specific values for \( A \) and \( B \) are given?
4. How would the solutions change if additional constraints are imposed on \( A \) and \( B \)?
5. Can we graph this equation to visualize the relationship between \( A \), \( B \), and \( C \)?

**Tip:** To check if solutions are real, examine the discriminant \((b^2 - 4ac)\). If it's non-negative, the roots are real; if negative, the roots are complex.

Learn how to solve the quadratic equation -44*A*2+12*AB*+8*B*2−7*AC*+*BC*−*C*2=0 with a detailed step-by-step solution. Understand the quadratic formula and how it applies to this specific problem involving variables A, B, and C. Suitable for advanced high school students and anyone interested in quadratic equations.

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