Problem 63 - Algebra 2

Question

Determine whether the value of \(c\) that makes \(a x^{2}+b x+c\) a perfect square trinomial is sometimes, always, or never negative. Explain your reasoning.

Step-by-Step Solution

Answer

c is never negative.

Step 1: Understand the Perfect Square Trinomial

A perfect square trinomial takes the form of \((ax + b)^2\). Expanding this, we get the expression \(a^2x^2 + 2abx + b^2\). Therefore, for \(ax^2 + bx + c\) to be a perfect square trinomial, it must match this form.

Step 2: Identify the Structure Required

To match the expanded form \((ax + b)^2\) with \(ax^2 + bx + c\), it must hold that \(a = a^2\), \(b = 2ab\), and \(c = b^2\). The focus here is on \(c\), which must be \(b^2\).

Step 3: Analyze the Value of c (b²)

Since \(b^2\) is a square of a real number \(b\), it is always non-negative. This is because squaring any real number, whether positive or negative, results in a non-negative number.

Step 4: Determine the Nature of c

Since \(c = b^2\) and the square of any real number cannot be negative, \(c\) is never negative.

Key Concepts

Square of a Real Number

A square of a real number is formed by multiplying the number by itself. This results in a new value known as the square. The squaring process always yields a non-negative result.
Let's consider two possible cases:

• When a real number is positive, multiplying it by itself yields a positive square.
• If the number is negative, its square would still be positive because two negative values multiplied together result in a positive value.

Therefore, regardless of whether a number is positive or negative, the result of its square is always non-negative. Hence, the expression involving the square of a real number means that its value cannot be below zero.

Trinomial Structure

A trinomial is a type of polynomial featuring three terms. Typically structured as \(ax^2 + bx + c\), it forms the basis for many algebraic problems. In the specific context of a perfect square trinomial, the structure follows a special format.
It appears as the square of a binomial, \((ax + b)^2\), which expands into \(a^2x^2 + 2abx + b^2\).

• First term: \(a^2x^2\) stems from the square of the first term in the binomial.
• Second Term: \(2abx\) comes from multiplying both terms of the binomial and then doubling the result.
• Third term: \(b^2\) is the square of the constant term in the binomial.

This specific structure helps in easily identifying and forming perfect squares, making complex expressions more manageable.

Non-negative Numbers

Non-negative numbers include all numbers that are greater than or equal to zero. This set of numbers includes zero itself and extends through all the positive values.
When analyzing algebraic expressions, especially those involving squares, understanding non-negative numbers is essential. This is because squares of real numbers, regardless of the original sign, are always non-negative.

• Zero multiplied by itself is zero, and therefore non-negative.
• Positive numbers naturally remain positive when squared.
• Negative numbers become positive when squared due to the rule of multiplying negatives.

So, when dealing with expressions where parameters like \(c\) in a polynomial are concerns of being negative or non-negative, this knowledge ensures \(c\) remains non-negative.