(ax-1)-x%5E2%2B4%3Dbx.jpeg "(4x+4)(ax-1)-x^2+4=bx")
Solving for a and b in the Equation (4x+4)(ax-1)-x^2+4=bx
This article explores how to solve for the unknown constants 'a' and 'b' in the equation:
(4x+4)(ax-1)-x^2+4=bx
Let's break down the steps to find the solution:
1. Expand the Equation
First, we need to expand the left-hand side of the equation by multiplying the factors:
(4x+4)(ax-1) = 4ax^2 - 4x + 4ax - 4
Combining like terms, we get:
4ax^2 + 4ax - 4x - 4 - x^2 + 4 = bx
2. Rearrange the Equation
Next, let's rearrange the equation to group the terms with similar powers of x:
(4a - 1)x^2 + (4a - 4)x = bx
3. Equate Coefficients
To make the equation true for all values of x, the coefficients of the corresponding powers of x on both sides of the equation must be equal. This gives us two equations:
- Equation 1: (4a - 1) = 0 (Coefficient of x^2)
- Equation 2: (4a - 4) = b (Coefficient of x)
4. Solve for a and b
Solving Equation 1 for 'a', we get:
4a = 1 a = 1/4
Substituting the value of 'a' into Equation 2, we get:
4(1/4) - 4 = b b = -3
Conclusion
Therefore, the values of a = 1/4 and b = -3 satisfy the given equation. This means that the equation (4x+4)(ax-1)-x^2+4=bx holds true when these values are substituted for 'a' and 'b'.