Problem 22 - Algebra 1: Concepts and Skills

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$ x^{2}+10 x $$

Step-by-Step Solution

Answer

The term 25 must be added to the expression \(x^{2} + 10x\) to make it a perfect square trinomial.

Step 1: Identifying the coefficient

Firstly, we identify the coefficient of x in our given equation. In our case \(x^{2} + 10x\), the coefficient of x is 10.

Step 2: Applying the formula for Perfect Square

Perfect square trinomial is of the form \( (a+b)^2=a^2+2ab+b^2\) where \(a=x\) and 2ab is the coefficient from our given polynomial. Here, we can identify that 'a' is 1 (from \(x^2\)), and '2ab' is 10.

Step 3: Finding the value of b

To find b, we'll use the equation \(2ab=10\). Substitute a=1 in the equation, we have \(2b=10\). Hence \(b=5\).

Step 4: Computing b square

Now we'll find the square of the number we got in the previous step. So, \(b^2=5^2=25\). This the term that needs to be added to the given equation to make it a perfect square trinomial.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations (like addition and subtraction). When looking at the expression x^2 + 10x, it's clear that it represents a polynomial, specifically a binomial since it has two terms. To manipulate this expression effectively, understanding its structure is crucial.

• x^2 represents the square of the variable x.
• The term 10x indicates that the variable x is being multiplied by 10.

Grasping this foundation helps when we move on to processes like factoring or completing the square, as you can recognize patterns that lead to these procedures. For instance, in converting a simple binomial into a perfect square trinomial, we seek to add a specific constant term, thus transforming the binomial into a trinomial with three terms, where the first and last terms are perfect squares and the middle term is twice the product of the roots of the first and last terms.

Factoring Trinomials

The process of factoring trinomials, especially the perfect square trinomials, is a fundamental technique in algebra. It allows us to express a polynomial as the product of two or more polynomials. For instance, if we create the perfect square trinomial from the expression x^2 + 10x, by adding the perfect square of half the coefficient of x, which is 25 in this case, we get x^2 + 10x + 25.

This trinomial can be factored into (x + 5)^2, which is its simplest form. When factoring, we look for patterns and use them to break down complex expressions into simpler, multiplicative components. Recognizing a perfect square trinomial is advantageous as it can be factored quickly, which is crucial for solving equations and simplifying expressions.

Completing the Square

The technique of completing the square involves transforming a quadratic expression into a perfect square trinomial, making it easier to solve quadratic equations. This method can also be useful in various applications such as finding the vertex of a parabola in the context of quadratic functions.

For the expression x^2 + 10x, we complete the square by adding and then factoring as demonstrated in the steps provided. This transforms our expression into (x + 5)^2. When we complete the square, we can also easily determine the maximum or minimum of the function, which is of high significance in graphing parabolas, optimization problems, and even in the derivation of the quadratic formula.

By mastering the steps to complete the square, students gain a powerful tool that simplifies the process of solving quadratic equations and unleashing the full potential of algebraic problem-solving.