How B And C Signs In Quadratic Equations Affect Signs In Factored Form

Understanding the intricate relationship between the coefficients of a quadratic equation and its factored form is a cornerstone of algebra. In this comprehensive exploration, we delve into the fascinating interplay between the signs of b and c in the standard form of a quadratic equation, x² + bx + c, and the resulting signs of p and q in its factored form, (x + p) (x + q). This knowledge empowers us to predict the nature of the roots and gain a deeper understanding of quadratic behavior. Let’s embark on this journey to unlock the hidden connections within these equations.

Decoding the Standard and Factored Forms of Quadratic Equations

Before we dive into the heart of the matter, let's solidify our understanding of the two key forms of quadratic equations:

• Standard Form: The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. For the purpose of this discussion, we'll focus on the simplified case where a = 1, resulting in the equation x² + bx + c = 0. The coefficient b represents the linear term, and c represents the constant term.
• Factored Form: The factored form of a quadratic equation is expressed as (x + p) (x + q) = 0, where p and q are constants representing the roots of the equation. This form reveals the values of x that make the equation true, which are x = -p and x = -q. Expanding the factored form, we get x² + (p + q) x + pq = 0. This expanded form allows us to directly compare the coefficients with the standard form.

The connection between these forms lies in the relationship between the coefficients. By equating the coefficients of the expanded factored form and the standard form, we observe that:

• b = p + q
• c = pq

These two equations are the key to unlocking the relationship between the signs of b, c, p, and q. Let's explore each case in detail.

Case 1: When c is Positive (c > 0)

When the constant term c is positive, it signifies that the product of p and q (pq) is positive. This crucial piece of information tells us that p and q must have the same sign. They are either both positive or both negative. This is because the product of two numbers is positive only if both numbers share the same sign. Understanding this fundamental principle is essential for deciphering the nature of the roots of the quadratic equation.

Now, let's delve deeper into the implications based on the sign of b:

Subcase 1.1: c is Positive and b is Positive (c > 0, b > 0)

In this scenario, we know that p and q have the same sign (from c > 0) and their sum (p + q) is positive (since b > 0). The only way for two numbers with the same sign to have a positive sum is if both p and q are positive. This means that the roots of the equation, which are -p and -q, will both be negative. Understanding this connection helps us visualize the parabola representing the quadratic equation. It will open upwards and intersect the x-axis at two negative points.

• Example: Consider the equation x² + 5x + 6 = 0. Here, b = 5 (positive) and c = 6 (positive). Factoring this equation, we get (x + 2)(x + 3) = 0. Both p = 2 and q = 3 are positive, confirming our rule. The roots are x = -2 and x = -3, both negative.

Subcase 1.2: c is Positive and b is Negative (c > 0, b < 0)

Here, p and q still have the same sign (c > 0), but their sum (p + q) is negative (b < 0). This implies that both p and q must be negative. Only two negative numbers can add up to a negative sum. Consequently, the roots of the equation, -p and -q, will both be positive. The parabola representing this equation will also open upwards but intersect the x-axis at two positive points. This understanding is crucial for solving quadratic inequalities and analyzing the behavior of quadratic functions.

• Example: Consider the equation x² - 5x + 6 = 0. Here, b = -5 (negative) and c = 6 (positive). Factoring this equation, we get (x - 2)(x - 3) = 0, which can be rewritten as (x + (-2))(x + (-3)) = 0. Both p = -2 and q = -3 are negative. The roots are x = 2 and x = 3, both positive.

Case 2: When c is Negative (c < 0)

When the constant term c is negative, it means the product of p and q (pq) is negative. This is a game-changer because it tells us that p and q must have opposite signs. One must be positive, and the other must be negative. This understanding is vital because it directly impacts the nature of the roots and the graph of the quadratic equation. The magnitude of b relative to p and q will determine which root is positive and which is negative.

Let's examine the implications based on the sign of b in this case:

Subcase 2.1: c is Negative and b is Positive (c < 0, b > 0)

In this scenario, we know that p and q have opposite signs, and their sum (p + q) is positive. This tells us that the positive number between p and q has a larger absolute value than the negative number. This is because, for the sum of two numbers with opposite signs to be positive, the positive number must