- Distribute the negative sign to the terms in the second polynomial: \(-(5 x^2 - 14 y^2 - 9 x) = -5 x^2 + 14 y^2 + 9 x\).
- Combine the original terms with the newly distributed terms: \(12 x^2 - 11 y^2 - 13 x - 5 x^2 + 14 y^2 + 9 x\).
- Group and add the coefficients of like terms: \((12-5)x^2 + (-11+14)y^2 + (-13+9)x\).
- The final simplified result is \(\boxed{7x^2+3y^2-4x}\).
Explanation
- Problem Analysis
We are tasked with finding the difference between two quadratic polynomials in terms of variables \(x\) and \(y\). The given expression is \((12 x^2 - 11 y^2 - 13 x) - (5 x^2 - 14 y^2 - 9 x)\). To simplify this, we must subtract the second polynomial from the first. This involves distributing the negative sign to each term within the second parenthesis and then grouping and combining like terms. Like terms are those with identical variable components, such as \(x^2\), \(y^2\), and \(x\).
- Simplifying the Expression
First, we distribute the negative sign across the terms inside the second set of parentheses:
\[-(5x^2 - 14y^2 - 9x) = -5x^2 + 14y^2 + 9x.\]
Now, we rewrite the expression by removing the parentheses:
\[12x^2 - 11y^2 - 13x - 5x^2 + 14y^2 + 9x.\]
Next, we group the like terms together:
\[(12x^2 - 5x^2) + (-11y^2 + 14y^2) + (-13x + 9x).\]
Finally, we perform the arithmetic for the coefficients of each group:
- For \(x^2\): \(12 - 5 = 7\), so we get \(7x^2\).
- For \(y^2\): \(-11 + 14 = 3\), so we get \(3y^2\).
- For \(x\): \(-13 + 9 = -4\), so we get \(-4x\).
Combining these, we obtain the simplified polynomial: \(7x^2 + 3y^2 - 4x\).
- Conclusion
Comparing our result, \(7x^2 + 3y^2 - 4x\), with the given options:
- \(7 x^2+3 y^2-4 x\)
- \(7 x^2-3 y^2-4 x\)
- \(7 x^2-25 y^2-22 x\)
- \(17 x^2-25 y^2-22 x\)
Our derived expression matches the first option. Thus, the correct difference of the polynomials is \(\boxed{7x^2+3y^2-4x}\).
Examples
In various fields like physics and engineering, you often need to subtract equations representing different forces or energy states to understand the net change. For example, if you are modeling the trajectory of a projectile, one polynomial might represent the total kinetic energy at one time, and another might represent the potential energy. Subtracting these polynomials allows you to calculate the net energy variation at any given point, helping to optimize system designs or predict motion more accurately. This type of algebraic manipulation is foundational for analyzing dynamic systems.