(x%2B5)%2Bax%3D(x-3)(x%2B12)%2Bb.jpeg "(x-5)(x+5)+ax=(x-3)(x+12)+b")
Solving for 'a' and 'b' in the Equation (x-5)(x+5)+ax=(x-3)(x+12)+b
This article will guide you through the process of solving for the unknown constants 'a' and 'b' in the given equation. We will utilize algebraic manipulation and simplification to arrive at the solutions.
Expanding and Simplifying the Equation
First, we need to expand the products on both sides of the equation:
- Left Side: (x-5)(x+5) + ax = x² - 25 + ax
- Right Side: (x-3)(x+12) + b = x² + 9x - 36 + b
Now, the equation becomes:
x² - 25 + ax = x² + 9x - 36 + b
Isolating 'a' and 'b'
To isolate the terms with 'a' and 'b', we can subtract x² from both sides:
-25 + ax = 9x - 36 + b
Next, we can rearrange the equation to group terms with 'a' and 'b' together:
ax - b = 9x - 11
Solving for 'a' and 'b'
Since the equation must hold true for all values of 'x', the coefficients of 'x' on both sides must be equal, and the constant terms must also be equal. This gives us two separate equations:
- Coefficient of x: a = 9
- Constant Term: -b = -11
Solving these equations, we get:
- a = 9
- b = 11
Conclusion
Therefore, the values of 'a' and 'b' that satisfy the equation (x-5)(x+5)+ax=(x-3)(x+12)+b are a = 9 and b = 11.