Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Find all integers \(c>0\) such that \(x^{2}+8 x+c\) can be factored. Describe how you found these values of \(c\).

Step-by-Step Solution

Answer

The integer value for \(c\) that allows the equation \(x^2+8x+c\) to be factored is \(16\)

Step 1: Identify the Coefficients

First, identify the coefficients in the quadratic equation. In the equation \(x^2+8x+c\), the coefficients are \(a=1\), \(b=4\), and \(c\). Check if the equation has the form of a perfect square trinomial \(a^2 +2ab + b^2\). Here, \(a=x\), \(2ab=8x\). So, \(b=4\). \(b^2\) should be equal to \(c\) for the equation to be a perfect square trinomial.

Step 2: Calculate the Value for C

Calculate the square of \(b\) to find the value of \(c\). This can be computed by squaring \(4\). Hence, \(c = 4^2 = 16\).

Step 3: Validate the result

Substitute \(c = 16\) to validate the result. The original function \(x^2 + 8x + c\) becomes \(x^2 + 8x + 16\) which can be factored into \((x+4)^2\). This confirms that \(c = 16\) is the value for which the equation can be factored.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. This means, if you have a trinomial of the form \(a^2 + 2ab + b^2\), it can be factored into \((a + b)^2\). Identifying this structure helps in simplifying quadratic expressions.

To determine if a trinomial is a perfect square, check if it adheres to the pattern:

• The first term \(a^2\) is a perfect square.
• The last term \(b^2\) is a perfect square.
• The middle term is precisely twice the product of the roots of the first and last terms.

In our exercise, the quadratic expression is \(x^2 + 8x + c\). We deduced that the middle term \(8x\) suggests a relationship where \(2ab = 8x\), leading to \(b = 4\). Thus, \(c = 4^2 = 16\) is the suitable value making it a perfect square trinomial.

Coefficients in Quadratics

Quadratic equations typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients. Each coefficient plays a distinct role in shaping the curve of the quadratic graph.

• The coefficient \(a\) determines the parabola's opening direction and width. A positive \(a\) ensures it opens upwards, while a negative \(a\) opens it downwards.
• The coefficient \(b\) influences the position of the parabola along the x-axis.
• The coefficient \(c\) represents the y-intercept, showing where the graph intersects the y-axis.

In the exercise, we have \(a = 1\), \(b = 8\), and \(c\). Finding \(c\) such that the expression can be a perfect square trinomial involves evaluating \(b^2 = c\). Here, \(b = 4\), thus \(c = 16\). Anchoring \(c\) to being a specific value ensures the quadratic fits the desired form.

Integer Solutions

Finding integer solutions in quadratic contexts often requires understanding the equation's factoring capabilities. For a quadratic expression to be easily factored, especially in terms of integer solutions, it may help to ensure it forms a perfect square trinomial when possible.

Integer solutions are usually targeted because they tend to simplify the solution process. Solutions usually involve integer coefficients that simplify the quadratic into multiple simpler binomials.

In our problem, after identifying that \(c = 16\), the quadratic \(x^2 + 8x + 16\) can be factored into \((x+4)^2\). This factoring confirms that solutions to the expression are integer-based and avoid complex roots, making it straightforward to evaluate further. Factoring into integers aligns with the goal of simplifying and deriving clear-cut solutions.