Problem 44 - Algebra 2

Question

Find the value of \(k\) that would make the left side of each equation a perfect square trinomial. $$ x^{2}-k x+81=0 $$

Step-by-Step Solution

Answer

The value of \(k\) that would make the left side of the equation a perfect square trinomial is \(k = 18\).

Step 1: Understanding a perfect square trinomial

A perfect square trinomial is of the form \(a^2 \pm 2ab + b^2\). It is the result of squaring a binomial \(a \pm b\). In this form, \(b^2\) is the last term and \(2ab\) is the middle term.

Step 2: Determine the last term of the binomial square

Given the equation \(x^2 - kx + 81 = 0\), the last term is 81. We know that 81 is a perfect square because \(81 = 9^2\). Therefore, in the binomial square \(x \pm b\), \(b^{2} = 81\) and \(b = 9\).

Step 3: Find the middle term of the binomial square

The middle term \(2ab\) in our case should be the \(kx\) term in the given equation. We have already determined that \(b = 9\). For \(a\), since we are dealing with \(x^{2}\), \(a\) is \(x\) or \(1\), therefore \(2ab = 2 \cdot 9 \cdot x\), which results in \(18x\).

Step 4: Set the middle term of the binomial square equal to the middle term of the given equation

We have found that the middle term from the perfect square would be \(18x\). Therefore, to make the left side of the given equation a perfect square trinomial, the value of \(k\) must be equal to the coefficient of the middle term we found: \(k = 18\).

Key Concepts

Algebraic Equations

Algebraic equations are the cornerstone of algebra, representing the relationship between variables and constants. An equation is a mathematical statement that asserts the equality of two expressions, often involving a variable, such as 'x', that we aim to solve for.

In the context of our problem, the algebraic equation is given by the pattern of a perfect square trinomial. A trinomial implies there are three terms, and in the case of a perfect square trinomial, these terms correspond to the squared binomial pattern, which we often encounter in algebraic manipulations. This framework allows us to seamlessly transform and manipulate equations to uncover the value of unknown elements, such as 'k' in our example.

Solving for 'k' in a perfect square trinomial equation requires us to understand binomial squares and how to factor quadratics, which are both critical skills in algebra.

Binomial Squares

Binomial squares are expressions obtained by squaring a binomial, which is an algebraic expression containing two terms joined by a plus or minus sign. When a binomial is squared, it results in a perfect square trinomial.

The general form of a squared binomial is \(a \pm b)^2 = a^2 \pm 2ab + b^2\). As shown in our step-by-step solution, identifying the terms of a squared binomial helps us find a perfect square trinomial. In the provided example, the equation \(x^2 - kx + 81 = 0\) is dissected to reveal that the constant term, 81, is actually the square of 9.

The knowledge of binomial squares is not only useful for identifying perfect square trinomials but also for simplifying complex algebraic expressions and equations. Recognizing the pattern early on can make the process of solving algebraic equations more intuitive.

Factoring Quadratics

Factoring quadratics is a method used to rewrite a quadratic expression as the product of two binomials. This technique is essential for solving quadratic equations, which are generally of the form \(ax^2 + bx + c = 0\). Factoring is particularly elegant when the quadratic is a perfect square trinomial because it simplifies to a squared binomial.

In the exercise presented, the trinomial \(x^2 - kx + 81\) hints at a hidden quadratic structure that can be expressed as a binomial square. To factor a perfect square trinomial, you need to determine the square root of the first and last terms and ensure that the middle term corresponds to twice the product of those roots, which is what we did to solve for 'k'.

A deep understanding of how to factor quadratics can greatly aid in the resolution of many algebraic problems, making it a pivotal technique for students to master. Recognizing patterns that lead to perfect square trinomials can simplify the otherwise complex process of solving quadratic equations.