How to Determine Perfect Square Trinomial
Determining whether a trinomial is a perfect square is an essential skill in algebra, as it helps in simplifying expressions and solving equations. A perfect square trinomial is a polynomial of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). To determine if a trinomial is a perfect square, follow these steps:
1. Identify the leading coefficient: The leading coefficient is the coefficient of the \( x^2 \) term. If it is not 1, divide the entire trinomial by the leading coefficient to make the leading coefficient equal to 1.
2. Check the middle term: The middle term, \( bx \), should be twice the product of the square root of the leading coefficient and the constant term. In other words, \( b = 2 \sqrt{a} \cdot c \). If this condition is met, the trinomial has a chance of being a perfect square.
3. Verify the constant term: The constant term, \( c \), should be the square of the middle term coefficient divided by the leading coefficient. If \( c = \left(\frac{b}{2\sqrt{a}}\right)^2 \), then the trinomial is a perfect square.
Let’s consider a few examples to illustrate these steps:
Example 1:
Given the trinomial \( 4x^2 + 12x + 9 \), we can see that the leading coefficient is 4, which is not 1. To make it 1, we divide the entire trinomial by 4:
\[ x^2 + 3x + \frac{9}{4} \]
Now, we check the middle term: \( 3 = 2 \sqrt{1} \cdot \frac{9}{4} \), which is true. Finally, we verify the constant term: \( \frac{9}{4} = \left(\frac{3}{2\sqrt{1}}\right)^2 \), which is also true. Therefore, \( 4x^2 + 12x + 9 \) is a perfect square trinomial.
Example 2:
Consider the trinomial \( x^2 – 6x + 9 \). The leading coefficient is 1, so we don’t need to make any adjustments. The middle term is \( -6x \), and the constant term is 9. Since \( -6 = 2 \sqrt{1} \cdot 3 \), and \( 9 = \left(\frac{-6}{2\sqrt{1}}\right)^2 \), we can conclude that \( x^2 – 6x + 9 \) is a perfect square trinomial.
By following these steps, you can determine whether a trinomial is a perfect square. This skill is not only useful in algebra but also in higher-level mathematics and problem-solving scenarios.
