Problem 76 - College Algebra and Calculus: An Applied Approach

Question

Find all integers \(b\) such that \(x^{2}+b x+24\) can be factored. Describe how you found these values of \(b\).

Step-by-Step Solution

Answer

The possible integer values of \(b\) for which the given equation \(x^{2}+b x+24\) can be factored are -10, -8, 8, and 10.

Step 1: Identify the coefficients

First, let's identify the coefficients of our quadratic equation. In this case, \(a = 1\), \(c = 24\) and \(b\) is the value we are asked to find.

Step 2: Formulate the condition for \(b\)

Recalling the discriminant formula \(b^{2}-4ac\), we substitute \(a = 1\) and \(c = 24\) into the formula. We want the discriminant to be a perfect square for the equation to be factorisable, so for some integer \(k\), we have \(b^{2} - 4*1*24 = k^{2}\).

Step 3: Solve for \(b\)

Rearranging the equation gives \(b^{2} = k^{2} + 4*1*24 = k^{2} + 96\). Since \(b\) is also an integer, it follows that \(b\) is the root of a perfect square plus 96. So, \(b\) can be any integer such that when squared and increased by 96, you get another square. By testing various perfect squares, we get \(b = -10, -8, 8, 10\).

Key Concepts

Using the Discriminant Method

The discriminant method is a powerful tool in solving and factoring quadratic equations. It's determined by the formula \(b^2 - 4ac\) when dealing with a quadratic like \(ax^2 + bx + c\).
In the context of factoring these types of equations, the discriminant reveals important information about the roots of the quadratic.

• If the discriminant is a perfect square, the quadratic can be factored over the integers.
• If it's positive but not a perfect square, the roots are real but irrational.
• If it’s zero, the quadratic has exactly one real distinct root.
• And if it’s negative, the quadratic has complex roots and cannot be factored over the real numbers.

For this exercise, to find the integer solutions of \(b\), we need \(b^2 - 4(1)(24) = b^2 - 96\) to be a perfect square, ensuring that factoring is possible.

Perfect Square Concept

A perfect square is simply an integer that is the square of another integer. For example, \(1, 4, 9, 16\), and so forth. These numbers are notable because when they appear in mathematical problems,they can simplify solutions and help identify integer factors easily.
In this scenario, because we needed \(b^2 - 96\) to be a perfect square, it implies that when you add or subtract 96 from some other squared integer \(k^2\), the resulting number \(b\) remains an integer.
Thus, knowing your perfect squares makes it straightforward to identify possible solutions for \(b\). By replacing \(b^2 = k^2 + 96\), if you test different values for \(k\), you can determine that \(k^2\) must be numbers like \(1, 4, 9, 16,\) etc., to satisfy the condition. By doing that, you'll find that \(b = -10, -8, 8,\) or \(10\).

Finding Integer Solutions

Integer solutions are solutions to equations where the answer is a whole number, without fractions or decimals. For the quadratic equation \(x^2 + bx + 24\), finding integer values of \(b\) that allow the equation to be factored relies heavily on the results of the discriminant method as explained before.
By setting \(b^2 - 96\) to equal a perfect square \(k^2\), we ensure our factored quadratic expressions \((x + m)(x + n)\) have integer coefficients. To find \(b\), assume \(b^2 - 96 = k^2\), thus rearranging gives \(b^2 = k^2 + 96\).

• Start by examining perfect squares near 96.
• Adjust \(k\) and check if \(b\) calculated is an integer.
• Possible values of \(k^2\) lead to integer values like -10, -8, 8, or 10.

Being systematic about checking each possible integer value of \(k\) helps confirm Iintegers that satisfy the equation. It’s crucial to understand that these integer solutions form because both the roots and the solutions of these equations have integer relationships when factored correctly.