(4x+4)(ax-1)-x^2+4=bx

2 min read Jun 16, 2024
(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'.

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