Problem 46 - Algebra and Trigonometry

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-\frac{1}{4} x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial \(x^{2} - \frac{1}{4}x\) to make it a perfect square trinomial is \frac{1}{64}. The factorized form of the perfect square trinomial is \(\left(x - \frac{1}{8}\right)^2\).

Step 1: Identify the coefficient of the linear term

For the equation \(x^{2} - \frac{1}{4}x \), the coefficient of the x-term (linear term) is - \frac{1}{4}.

Step 2: Compute the square of half of the linear term’s coefficient

Half of - \frac{1}{4} is - \frac{1}{8}. The square of - \frac{1}{8} is \( \left(- \frac{1}{8}\right)^2 = \frac{1}{64} \)

Step 3: Add computed value to the quadratic expression

Adding \frac{1}{64} to the original quadratic expression gives \(x^{2} - \frac{1}{4}x + \frac{1}{64} \)

Step 4: Factorize the perfect square trinomial

The perfect square trinomial \(x^{2} - \frac{1}{4}x + \frac{1}{64} \) is the square of a binomial. It factorizes to \(\left(x - \frac{1}{8}\right)^2 \)

Key Concepts

Factoring Trinomials

Factoring trinomials, particularly quadratic trinomials, is a foundational skill in algebra that enables us to simplify expressions and solve quadratic equations. A trinomial is a polynomial with three terms, commonly written in the form of \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) represent constants. The process of factoring converts this trinomial into a product of two binomials.

When we factor, we look for two numbers that multiply to give the constant term, \(c\), and add to give the middle coefficient, \(b\). But not all trinomials are immediately factorable. In some cases, you might need to use the technique for creating a perfect square trinomial, which involves completing the square.

This technique is particularly handy when the trinomial does not have readily apparent factors, or in the case of our original exercise, where we need to add a specific constant to make it a perfect square. The added constant is the square of half the coefficient of \(x\), which ensures we get a binomial square after factoring the trinomial.

Binomial Squares

A binomial square arises from squaring a binomial expression, and it results in a specific form of a quadratic trinomial known as a perfect square trinomial. A binomial is an algebraic expression containing two terms. For example, squaring the binomial \(x + y\) gives \(x^2 + 2xy + y^2\), which is a perfect square trinomial because it can be expressed as \(x + y\)^2.

To construct a perfect square trinomial from a binomial, we need to follow these steps:

• Take the square of the first term.
• Take the square of the second term.
• Add twice the product of the first and second terms.

This formula comes from the identity \(\(a+b\))^2 = a^2 + 2ab + b^2\). In the exercise, identifying the constant to add to a binomial to create a perfect square trinomial follows from performing the reverse operation: we figure out what constant will complete the square.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, which generally assume the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients and \(a\) is not zero. These expressions graphically represent a parabolic curve on a coordinate plane.

Working with quadratic expressions often involves factoring them into binomials or completing the square to transform them into a perfect square trinomial, which is a specific form of a quadratic that factors into a squared binomial. The elegance of perfect square trinomials lies in their symmetry and predictability, making them easier to factor and solve than most generic quadratic expressions.

As demonstrated in the exercise, completing the square to form a perfect square trinomial is a practical method for solving quadratics. It can also aid in the derivation of the quadratic formula, a universal method for finding the roots of any quadratic expression. By mastering the transformation of quadratic expressions into perfect square trinomials, students gain a powerful tool for algebraic manipulation and problem-solving.