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

Jasonbradley

3 min read

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Jun 17, 2024

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Factoring and Solving (x-1)(2x+3)(2x-3)

This expression is already factored, but we can explore its properties and how to solve for its roots.

Understanding the Expression

• Factored Form: The expression is presented in factored form, meaning it's already broken down into simpler expressions multiplied together.
• Factors: The factors are:
  • (x-1)
  • (2x+3)
  • (2x-3)
• Roots: The roots of an expression are the values of x that make the expression equal to zero. To find the roots, we set each factor equal to zero and solve:

Finding the Roots

1. (x-1) = 0 Solving for x, we get x = 1.

2. (2x+3) = 0 Solving for x, we get x = -3/2.

3. (2x-3) = 0 Solving for x, we get x = 3/2.

Conclusion

Therefore, the roots of the expression (x-1)(2x+3)(2x-3) are x = 1, x = -3/2, and x = 3/2. This means the expression equals zero when x takes on these values.

Expanding the Expression (Optional)

If we want to see what the expression looks like in its expanded form, we can multiply out the factors:

1. (2x+3)(2x-3): This is a difference of squares pattern: (a+b)(a-b) = a² - b². Therefore: (2x+3)(2x-3) = (2x)² - (3)² = 4x² - 9

2. (x-1)(4x² - 9): Now we multiply this by the remaining factor: (x-1)(4x² - 9) = 4x³ - 9x - 4x² + 9 = 4x³ - 4x² - 9x + 9

The expanded form of the expression is 4x³ - 4x² - 9x + 9. This is equivalent to the factored form, but it can be useful for certain operations like finding the y-intercept.