What value of c will make a perfect-square trinomial?
Perfect-square trinomials are a fundamental concept in algebra, representing expressions that can be factored into the square of a binomial. Understanding the value of c that makes a trinomial a perfect square is crucial for simplifying expressions, solving equations, and exploring various mathematical properties. In this article, we will delve into the characteristics of perfect-square trinomials and identify the specific value of c that ensures a trinomial is a perfect square.
A perfect-square trinomial has the form \(a^2 + 2ab + b^2\), where \(a\) and \(b\) are real numbers. This form is derived from the expansion of the binomial \((a + b)^2\). The middle term, \(2ab\), is the product of twice the product of \(a\) and \(b\). To determine the value of \(c\) that makes a trinomial a perfect square, we need to analyze the relationship between the coefficients of the trinomial.
Consider a general trinomial \(ax^2 + bx + c\). For this trinomial to be a perfect square, it must be of the form \(a(x + d)^2\), where \(d\) is a real number. Expanding this expression, we get \(a(x^2 + 2dx + d^2)\). Comparing this with the general trinomial, we can see that:
1. The coefficient of \(x^2\) in the perfect square is \(a\), which is the same as the coefficient of \(x^2\) in the general trinomial.
2. The coefficient of \(x\) in the perfect square is \(2ad\), which is equal to \(b\) in the general trinomial.
3. The constant term in the perfect square is \(ad^2\), which is equal to \(c\) in the general trinomial.
From these comparisons, we can derive the following relationships:
1. \(a = a\), which is trivially true.
2. \(2ad = b\), which implies \(d = \frac{b}{2a}\).
3. \(ad^2 = c\), which can be rewritten as \(c = a\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a}\).
Thus, for a trinomial \(ax^2 + bx + c\) to be a perfect square, the value of \(c\) must be \(\frac{b^2}{4a}\). This value ensures that the trinomial can be factored into the square of a binomial, simplifying algebraic expressions and enabling further mathematical exploration.
