Problem 34 - Elementary Algebra

Question

Factor. $$ 54 x 2-15 x+6 $$

Step-by-Step Solution

Answer

The factored form is \(3(2x - 1)(9x + 2)\).

Step 1: Identify the Expression

The expression given to us is \( 54x^2 - 15x + 6 \). Our objective is to factor this quadratic expression.

Step 2: Identify a Common Factor

Examine each term to see if there is a common factor. In this case, 3 is a common factor for all terms. Factoring 3 out gives us: \( 3(18x^2 - 5x + 2) \).

Step 3: Factor the Quadratic Expression

Now, focus on factoring the quadratic expression \( 18x^2 - 5x + 2 \). We need two numbers whose product is \(18 \times 2 = 36\) and sum is \(-5\). The numbers are -9 and 4.

Step 4: Rewrite the Middle Term

Rewrite the middle term \(-5x\) using the numbers found in Step 3. The expression becomes \(18x^2 - 9x + 4x + 2\).

Step 5: Group and Factor by Grouping

Group the terms: \((18x^2 - 9x) + (4x + 2)\). Factor out the greatest common factor in each group to get: \(9x(2x - 1) + 2(2x - 1)\).

Step 6: Factor the Common Binomial Factor

Notice that \((2x - 1)\) is common in both groups. Factor out \((2x - 1)\) to get: \((2x - 1)(9x + 2)\).

Step 7: Combine with the Common Factor from Step 2

Don't forget the common factor of 3 that we factored out in Step 2. The fully factored form of the expression is: \(3(2x - 1)(9x + 2)\).

Key Concepts

Common Factors

When we talk about common factors in mathematics, we're referring to a number or algebraic term that can evenly divide all terms in a given expression. In the expression \(54x^2 - 15x + 6\), it's essential to first look for common factors to simplify our work. A common factor makes an expression easier to manage by reducing it to smaller, more workable parts.

Let's break this down:

• Scan through all the coefficients and identify any common divisors.
• For \(54, -15,\) and \(6\), you might notice the smallest positive number dividing them is \(3\).
• By factoring out \(3\), we divide each term by \(3\), simplifying the original expression to \(3(18x^2 - 5x + 2)\).

Finding common factors is often the first and crucial step in factoring quadratic expressions, as it helps break down the problem into smaller, more manageable pieces.

Quadratic Expressions

Quadratic expressions are mathematical expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These expressions describe parabolic shapes when graphed and are fundamental in algebra.

For the given expression \(18x^2 - 5x + 2\), let's analyze:

• \(a = 18\), which affects how "steep" or "wide" the parabola is.
• \(b = -5\), a parameter that influences the position of the axis of symmetry and the expression's linear component.
• \(c = 2\), a constant impacting the vertical positioning of the parabola on the graph.

Understanding the structure of quadratics helps us decide the best factoring method, like looking for two numbers that multiply to \(ac\) (here \(18 \times 2 = 36\)) but add up to \(b\) (\(-5\)). This step assists us in reconfiguring terms for further simplification.

Factoring by Grouping

Once a quadratic expression is simplified with common factors, we employ factoring by grouping to break it down further. This technique involves reorganizing terms to reveal further common factors.

To use this strategy in \(18x^2 - 5x + 2\), we:

• Seek two numbers that multiply to \(36\) and add to \(-5\). Here, those numbers are \(-9\) and \(4\).
• Rewrite the middle term, \(-5x\), using these two numbers, yielding \(18x^2 - 9x + 4x + 2\).
• Group terms: \((18x^2 - 9x) + (4x + 2)\).
• Factor out common terms in each group: \(9x(2x - 1) + 2(2x - 1)\).
• Notice the binomial \((2x - 1)\) is common; factor it out to achieve \((2x - 1)(9x + 2)\).

The last step involves using these common binomials to piece together the expression into multiplied components, along with the previously factored \(3\), resulting in the fully factored form \(3(2x - 1)(9x + 2)\).