(x+1)(4x-2)=(2x-1)(3x-1)

Jasonbradley

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Jun 16, 2024

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

This article will guide you through solving the equation (x+1)(4x-2) = (2x-1)(3x-1).

Expanding Both Sides

The first step is to expand both sides of the equation using the distributive property (also known as FOIL).

• Left Side: (x+1)(4x-2) = 4x² - 2x + 4x - 2 = 4x² + 2x - 2
• Right Side: (2x-1)(3x-1) = 6x² - 2x - 3x + 1 = 6x² - 5x + 1

Now, our equation looks like this: 4x² + 2x - 2 = 6x² - 5x + 1

Simplifying the Equation

To make the equation easier to solve, we'll move all the terms to one side:

• Subtract 4x² from both sides: 2x - 2 = 2x² - 5x + 1
• Add 5x to both sides: 7x - 2 = 2x² + 1
• Subtract 1 from both sides: 7x - 3 = 2x²

Now we have a quadratic equation: 2x² - 7x + 3 = 0

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Where:

• a = 2
• b = -7
• c = 3

Plugging these values into the quadratic formula:

x = (7 ± √((-7)² - 4 * 2 * 3)) / (2 * 2)

x = (7 ± √(49 - 24)) / 4

x = (7 ± √25) / 4

x = (7 ± 5) / 4

This gives us two possible solutions:

• x = (7 + 5) / 4 = 3
• x = (7 - 5) / 4 = 1/2

Checking the Solutions

It's always a good idea to check our solutions by plugging them back into the original equation.

• For x = 3:

  • (3+1)(43-2) = (23-1)(3*3-1)
  • 4 * 10 = 5 * 8
  • 40 = 40 (This solution works)

• For x = 1/2:

  • (1/2+1)(4*(1/2)-2) = (2*(1/2)-1)(3*(1/2)-1)
  • (3/2) * 0 = 0 * (1/2)
  • 0 = 0 (This solution also works)

Conclusion

Therefore, the solutions to the equation (x+1)(4x-2) = (2x-1)(3x-1) are x = 3 and x = 1/2.