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(2x+1)+3/2x+1=4 X≠-1/2

3 min read Jun 16, 2024
(2x+1)+3/2x+1=4 X≠-1/2

Solving the Equation (2x+1)+3/(2x+1)=4

This article will guide you through solving the equation (2x+1)+3/(2x+1)=4, with the restriction that x≠-1/2.

1. Understanding the Equation

The equation involves a rational expression (a fraction with variables in the denominator). The restriction x≠-1/2 is crucial because it prevents the denominator from becoming zero, which would make the expression undefined.

2. Simplifying the Equation

To solve the equation, we need to manipulate it to isolate x. Let's start by combining the terms on the left side:

  • (2x+1)+3/(2x+1) = 4
  • [(2x+1)² + 3]/(2x+1) = 4

3. Eliminating the Fraction

Multiply both sides of the equation by (2x+1) to eliminate the fraction:

  • [(2x+1)² + 3]/(2x+1) * (2x+1) = 4 * (2x+1)
  • (2x+1)² + 3 = 8x + 4

4. Expanding and Rearranging

Expand the left side of the equation and rearrange terms to get a quadratic equation:

  • 4x² + 4x + 1 + 3 = 8x + 4
  • 4x² - 4x = 0

5. Solving the Quadratic Equation

Factor out a 4x from the equation:

  • 4x(x - 1) = 0

This gives us two possible solutions:

  • x = 0
  • x = 1

6. Verification

Since we had a restriction that x≠-1/2, both solutions are valid. To verify, substitute each solution back into the original equation and check if it holds true.

Conclusion

The equation (2x+1)+3/(2x+1)=4 has two solutions: x=0 and x=1, both satisfying the restriction x≠-1/2.

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