How do I factorise the expression 45x^2 + 60x + 220x?

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Factoring the Expression 45x² + 60x + 220

To factor the expression 45x² + 60x + 220, we will follow a systematic approach:

Step 1: Identify the Expression

The expression is polynomial in the form of ax² + bx + c, where:

  • a = 45
  • b = 60
  • c = 220

Step 2: Look for a Common Factor

First, we will check if there’s a common factor among the coefficients of the terms. The coefficients are 45, 60, and 220. The greatest common divisor (GCD) of these numbers is 15.

Now, we can factor out 15 from the expression:

45x² + 60x + 220 = 15(3x² + 4x + rac{220}{15})

Now simplifying:

15(3x² + 4x + 14.67)

Step 3: Factor the Quadratic Inside

The next step is to focus on the quadratic 3x² + 4x + 14.67. However, it doesn’t factor neatly with integers, so we will use the quadratic formula:

x = (-b ± √(b²-4ac)) / 2a

Here, a = 3, b = 4, and c = 14.67. Plugging in these values:

x = (-4 ± √(4² - 4 × 3 × 14.67)) / (2 × 3)
	  = (-4 ± √(16 - 176.04)) / 6
	  = (-4 ± √(-160.04)) / 6

This indicates that there are complex roots, and it suggests that the quadratic does not have simple factorization. Therefore, the expression will remain as:

45x² + 60x + 220 = 15(3x² + 4x + 14.67)

Conclusion

The final factored answer of the expression is:

15(3x² + 4x + 14.67)

If you’re looking for integer factorization, this polynomial does not factor into integers due to the nature of its roots.

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