Problem 38 - Beginning and Intermediate Algebra

Question

Factor by trial and error. $$6 t^{2}+23 t+7$$

Step-by-Step Solution

Answer

The factored form of the quadratic equation \(6t^2 + 23t + 7\) by trial and error is \((2t + 7)(3t + 1)\).

Step 1: Identify the Coefficients

Identify the coefficients of the quadratic equation, which are 6, 23, and 7 in this case. Write the equation as: \[6t^2 + 23t + 7\]

Step 2: Factor the Coefficients

List the factors of the first coefficient (6) and the last coefficient (7). Factors of 6: (1, 6), (2, 3) Factors of 7: (1, 7)

Step 3: Test the Combinations of Factors

Use the factors to create binomial combinations and check if their product equals the given quadratic equation. Try combining (1, 6) and (1, 7): \[(t + 1)(6t + 7)\] This product results in \(6t^2 + 13t + 7t + 7\), which simplifies to \(6t^2 + 20t + 7\). This combination doesn't equal the given equation. Next, try combining (2, 3) and (1, 7): \[(2t + 1)(3t + 7)\] This product results in \(6t^2 + 14t + 3t + 7\), which simplifies to \(6t^2 + 17t + 7\). This combination doesn't equal the given equation either. Lastly, try switching the positions of the factors of 7: \[(2t + 7)(3t + 1)\] This product results in \(6t^2 + 2t + 21t + 7\), which simplifies to \(6t^2 + 23t + 7\). This combination equals the given equation.

Step 4: Write the Factored Form

Since the combination \((2t+7)(3t+1)\) results in the given equation, the factored form of \(6t^2 + 23t + 7\) by trial and error is: \[(2t + 7)(3t + 1)\]

Key Concepts

Trial and Error Method

The trial and error method for factoring quadratics might sound a bit random, but it's a systematic approach to zeroing in on the right factors of a quadratic expression. The goal is to find two binomials whose product is the original quadratic equation.
This involves selecting different combinations of factors from the quadratic coefficients and testing these until the desired results are achieved.

• Select potential factors of the leading and trailing coefficients.
• Create binomial expressions from these factor pairs.
• Multiply the binomials to check if they expand to the original quadratic equation.

This method might require persistence and practice, as the right combination might not be evident from the outset. With experience, however, you'll be better positioned to make educated guesses, reducing the time involved.

Quadratic Coefficients

In a quadratic equation of the form ax² + bx + c, the numbers a, b, and c are known as the coefficients. For the given exercise, these coefficients are 6, 23, and 7, respectively.
Identifying these coefficients is the first step in the trial and error method.

• a (leading coefficient): Represents the number multiplying the squared term, which impacts the width and direction of the parabola.
• b (linear coefficient): Directly influences the shape and position of the curve.
• c (constant term): Provides the vertical intercept of the quadratic on the graph.

Understanding these coefficients helps in selecting appropriate factor pairs, which, when tried and tested, yield the original equation upon expansion.

Binomial Combinations

The process of factoring quadratics revolves heavily around creating and testing binomial combinations. These combinations are constructed using the factors of the quadratic coefficients.
In our exercise, potential binomial combinations were created using the factors (1, 6) and (2, 3) for the leading coefficient 6, and (1, 7) for the constant term 7.
Here's how it works:

• Create pairs from the factors identified.
• Build binomials by associating these pairings with the variables.
• Expand the binomials to confirm that the product replicates the original quadratic.

This technique may involve several attempts with different factor pairs, necessitating patience to achieve an exact match with the original equation.

Factored Form

Once the correct binomial combination is found, expressing the quadratic in its factored form becomes straightforward. In our example, through trial and error, we discovered that o(2t + 7)(3t + 1) o is the perfect factorization of the quadratic equation 6t² + 23t + 7.

• Why is Factored Form Important? It illustrates the roots of the quadratic equation and simplifies solving for the variable t.
• Offers a direct way to find the x-intercepts of the quadratic when graphed, revealing points where it crosses the x-axis.
• Provides insight into the structure and properties of the quadratic function.

The factored form is essentially the condensed version of a quadratic expression, lending a clearer view of its characteristics and facilitating easier manipulation in algebraic operations.