Problem 24 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ 6 x^{2}-13 x+2 $$

Step-by-Step Solution

Answer

The expression factors to \((x - 2)(6x - 1)\).

Step 1: Factor Coefficient Matching

To write the expression as a product of two binomials, begin by looking at the quadratic form: \(ax^2 + bx + c\). Here, \(a = 6\), \(b = -13\), and \(c = 2\). We need to find two numbers that multiply to \(a imes c = 12\) and add to \(-13\). These numbers are \(-12\) and \(-1\).

Step 2: Rewrite the Middle Term

Rewrite the middle term \(-13x\) using the numbers found: \(-12\) and \(-1\). The expression becomes: \(6x^2 - 12x - 1x + 2\).

Step 3: Factor by Grouping

Separate the expression into two groups: \((6x^2 - 12x)\) and \((-1x + 2)\). Factor out the greatest common factor from each group. From \(6x^2 - 12x\), factor out \(6x\): \(6x(x - 2)\). From \(-1x + 2\), factor out \(-1\): \(-1(x - 2)\).

Step 4: Identify Common Binomial

Notice that \((x - 2)\) is a common binomial in both terms. Factor out \((x - 2)\), giving you \((x - 2)(6x - 1)\).

Step 5: Write the Final Factored Form

The expression \(6x^2 - 13x + 2\) is now written as the product of two binomials: \((x - 2)(6x - 1)\).

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Understanding quadratic equations is crucial because they appear in various mathematical contexts and applications, such as physics, engineering, and economics.

Quadratic equations have a characteristic U-shaped graph known as a parabola. The highest or the lowest point of the parabola is called the vertex. The solutions to a quadratic equation, also known as the roots, are the values of \(x\) at which the parabola intersects the x-axis. These roots can be real or complex, depending on the discriminant \(b^2 - 4ac\).

One way to solve quadratic equations is by factoring them into products of binomials. This method is effective when the equation can be expressed in a form that readily splits into binomial pairs.

Binomials

A binomial is an algebraic expression that contains exactly two terms. Examples of binomials include expressions like \(x + 3\) or \(2x - 5\). In factoring, binomials are key because they are often the building blocks we use to rewrite quadratic expressions into simpler multiplication forms.

When a quadratic equation is expressed as the product of two binomials, it can often lead to solving the equation more efficiently. The beauty of binomials in this context is their simplicity, making complex expressions appear much more manageable.

In practice, finding the right pair of binomials involves identifying numbers that contribute both to the final term (the constant after multiplication) and appropriately adjust the middle term, thus helping to break down the quadratic expression into its factored form. This decomposition is evident when turning \(6x^2 - 13x + 2\) into \((x - 2)(6x - 1)\).

Factoring by Grouping

Factoring by grouping is a technique used to break down polynomials into simpler expressions and is especially handy for quadratic expressions that are not immediately factorable. This method requires setting up the polynomial in pairs and identifying common factors from each pair.

To apply factoring by grouping, start by rewriting the quadratic expression in such a way that its middle term is split into two terms. For example, the expression \(6x^2 - 13x + 2\) is first rewritten as \(6x^2 - 12x - 1x + 2\).

Once you have this breakdown, group the terms: \((6x^2 - 12x)\) and \((-1x + 2)\). Then, factor out the greatest common factors from each group. In this case, \(6x\) is factored from the first group, and \(-1\) from the second. Combining these factors with the common binomial factor gives the factored form: \((x - 2)(6x - 1)\).

This method is powerful because it transforms expressions using a systematic approach, making it easier to solve quadratic equations without guessing.