Problem 143 - College Algebra

Question

Find all integers b so that the trinomial can be factored. $$ x^{2}+4 x+b $$

Step-by-Step Solution

Answer

The integer value of \( b \) for which the trinomial \( x^{2} + 4x + b \) can be factored is 3.

Step 1: Define the discriminant

First thing, that needs to be remembered is that the discriminant is defined as \( D = b^2 - 4ac \). In this exercise, \( a = 1 \), \( b = 4 \) and \( c \) is the unknown \( b \). Thus, the discriminant becomes \( D = b = 16 - 4b \).

Step 2: Solve for b

In order for the quadratic to be factorable, the discriminant (\( D \)) should be a perfect square. So, solve the equation \( D = 16 - 4b \) for \( b \) by first rearranging this equation to \( 4b = 16 - D \). The distance from \( 16 \) (which is 4 squared) to the nearest perfect squares (9 and 25) is 7 and 9. Therefore, solve for \( b \) using the equation \( 4b = 16 ±N \), where \( N \) can be 7 or 9. The solutions are \( b = 3 \) or \( b = -2 \)

Step 3: Check the solutions

Check if the solutions for \( b \) are valid by checking if the trinomials \( x^2 + 4x + 3 \) and \( x^2 + 4x - 2 \) can be factored. Indeed, the former one can be factored into \( (x + 1)(x + 3) \), while the latter can't be factored over integers. Thus, the only integer solution for \( b \) is 3.

Key Concepts

Discriminant in Quadratic Equations

When dealing with quadratic equations, understanding the role of the discriminant is crucial. The discriminant is part of the quadratic formula and is defined as \( D = b^2 - 4ac \). It helps determine the nature of the roots of the quadratic equation. In simple terms, it tells us whether the roots are real, complex, or repeated.
In the exercise given, we are considering the quadratic equation \( x^2 + 4x + b \). Here, \( a = 1 \), \( b = 4 \), and \( c \) is the unknown value \( b \). Therefore, the discriminant formula turns into \( D = 16 - 4b \).
The value of the discriminant can indicate:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one repeated real root.
• If \( D < 0 \), the equation has no real roots (it has two complex roots).

Determining \( D \) helps us decide whether the quadratic can be factored over real numbers.

Perfect Square

In the process of solving quadratic equations, identifying perfect squares can simplify factoring. A perfect square is a number that can be expressed as the square of an integer. For instance, numbers like 9, 16, 25 are perfect squares because they are the squares of 3, 4, and 5 respectively.

For a quadratic trinomial to be factorable with integer solutions, the discriminant should be a perfect square. In our exercise, we found that the potential value for the discriminant \( D = 16 - 4b \) needs to be set as a perfect square. By effectively adjusting \( b \) so that \( 16 - 4b \) becomes a perfect square, the trinomial \( x^2 + 4x + b \) becomes factorable.

• Perfect squares near 16 include 9 (\( 3^2 \)) and 25 (\( 5^2 \)).
• This insight helps us solve \( D = 16 - 4b \) using integers by setting it equal to these perfect squares.

This approach ensures that the quadratic equation is factorable in the integer domain.

Integer Solutions

For the given exercise, finding integer solutions for \( b \) is an essential goal. An integer solution means that \( b \) is a whole number that makes the trinomial \( x^2 + 4x + b \) factorable into linear factors with integer coefficients.

Having integer solutions eases the computational effort since it allows the use of basic arithmetic without involving complex numbers or fractions. The solved exercise showed that only \( b = 3 \) worked because this value ensures that the equation becomes \( x^2 + 4x + 3 \), which factors neatly into \( (x + 1)(x + 3) \), both having integer coefficients.

• This highlighted that to remain factorable over the integers, the numerical solutions to \( D = 16 - 4b \) had to meet specific criteria of being linked to perfect squares, therefore simplifying checks on results.

This method illuminates the logical steps necessary for simplifying quadratic trinomials into factors with integer solutions.

Quadratic Factoring

Quadratic factoring is a powerful technique used to break down a quadratic expression into the product of two linear factors. This method is particularly useful when you need to solve quadratic equations without using the quadratic formula.

To factor a quadratic trinomial such as \( x^2 + 4x + b \), you need to ensure it can be expressed as \( (x + m)(x + n) \), where \( m \) and \( n \) are integers. The key here is ensuring that the expression becomes factorable by appropriately choosing \( b \) to satisfy this condition. This means \( b \) should be adjusted so the middle term (\( 4x \) in our case) is a result of \( m + n = 4 \), and the constant term \( b \) is \( mn \).

• In the example, we tried \( b = 3 \), which worked as it allowed the quadratic to be factored as \( (x + 1)(x + 3) \).
• This is a prime example of trial and error aligned with logical reasoning, narrowing down the integers that make the trinomial factorable over the integers.

The technique highlights the essence of systematically evaluating potential solutions for their validity in ensuring factors have integer coefficients.