Verified Answer :

  • The expression \(16x^2 - 8x + 1\) is a perfect square trinomial, factoring to \(\boxed{(4x - 1)^2}\).
  • The expression \(x^2 - x - 30\) is a quadratic trinomial, factoring to \(\boxed{(x - 6)(x + 5)}\).
  • The expression \(9x^2 - 49\) is a difference of squares, factoring to \(\boxed{(3x - 7)(3x + 7)}\).
  • The expression \(3x^2 + 17x - 6\) is factored using the grouping method to \(\boxed{(3x - 1)(x + 6)}\).

Explanation

  1. Problem Analysis

We are tasked with factoring four different quadratic expressions. Each expression has its own specific structure that allows for a systematic approach to factoring. We will analyze each one using known algebraic patterns.

  1. Factoring 16x^2 - 8x + 1

The first expression is \(16x^2 - 8x + 1\). We recognize this as a perfect square trinomial of the form \(a^2x^2 - 2abx + b^2\), which factors as \((ax - b)^2\). Here, \((4x)^2 - 2(4x)(1) + 1^2 = (4x - 1)^2\). Thus, \(16x^2 - 8x + 1 = \boxed{(4x - 1)^2}\).

  1. Factoring x^2 - x - 30

The second expression is \(x^2 - x - 30\). We look for two numbers that multiply to \(-30\) and add up to the coefficient of \(x\), which is \(-1\). These two numbers are \(-6\) and \(5\), since \((-6) \times 5 = -30\) and \(-6 + 5 = -1\). Therefore, we can write the expression as \((x - 6)(x + 5)\). So, \(x^2 - x - 30 = \boxed{(x - 6)(x + 5)}\).

  1. Factoring 9x^2 - 49

The third expression is \(9x^2 - 49\). This is a difference of squares, which follows the pattern \(a^2 - b^2 = (a - b)(a + b)\). Identifying the squares, we have \((3x)^2 - 7^2\). This factors as \((3x - 7)(3x + 7)\). So, \(9x^2 - 49 = \boxed{(3x - 7)(3x + 7)}\).

  1. Factoring 3x^2 + 17x - 6

The fourth expression is \(3x^2 + 17x - 6\). We use the ac-method, where we look for two numbers that multiply to \(a \times c = 3 \times (-6) = -18\) and add up to \(b = 17\). These numbers are \(18\) and \(-1\). We split the middle term: \(3x^2 + 18x - x - 6\). Now we factor by grouping: \(3x(x + 6) - 1(x + 6)\), which simplifies to \((3x - 1)(x + 6)\). So, \(3x^2 + 17x - 6 = \boxed{(3x - 1)(x + 6)}\).

Examples

Factoring quadratic expressions is a fundamental skill in algebra with many real-world applications. For example, if you are designing a rectangular garden with an area represented by the quadratic expression \(x^2 + 5x + 6\), factoring it into \((x + 2)(x + 3)\) allows you to determine that the length and width of the garden are \(x+2\) and \(x+3\). This simplifies calculations for things like finding the perimeter or the cost of fencing, which is essential in fields like landscape architecture and urban planning.

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