Problem 3 - Algebra 1

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}+20 x$$

Step-by-Step Solution

Answer

The term that should be added to the expression to create a perfect square trinomial is 100.

Step 1: Identify the structure of a perfect square trinomial

A perfect square trinomial is of the form \(a^{2}+2ab+b^{2}\). It can be made by squaring a binomial (a+b)^{2} or (a-b)^{2}.

Step 2: Write the given expression as a uncompleted square trinomial

We have \(x^{2}+20x\). Our aim is to find the term which, when added to this expression, will complete the square. This will be equivalent to finding the 'b^{2}' term in our perfect square trinomial. The 2ab term is equivalent to 20x. Therefore, by substituting x for a and comparing terms, we see that 2ab = 20x which simplifies to 2b = 20.

Step 3: Calculate the value of b

Solving our equation for b gives us b = 20/2 = 10.

Step 4: Calculate the perfect square term

Now that we have the value of 'b', we can find the perfect square term 'b^{2}', which is going to be 10^{2} = 100 which will be added to the expression to make it a perfect square trinomial.

Step 5: Validate the perfect square trinomial

Insert the term into our original binomial to confirm that we have created a perfect square trinomial: \(x^{2} + 20x + 100\) is indeed \((x+10)^{2}\), a perfect square trinomial.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique makes equations easier to solve, especially when dealing with quadratic equations. The aim is to add a specific constant term to a quadratic expression so that it becomes a complete square. For example, consider the expression: \(x^2 + 20x\). To complete the square, you need to find a term that turns it into a perfect square trinomial. When you have an expression of the form \(x^2 + 2bx\), the term to be added is \(b^2\). By rearranging, you solve for 'b' using the equation \(2b = \) coefficient of \(x\). Here, it is 20, leading to \(b = 10\). Adding \(b^2 = 100\) to \(x^2 + 20x\) completes the square, resulting in the expression \(x^2 + 20x + 100 = (x + 10)^2\). This expression is now a perfect square trinomial, making it much simpler to handle during problem solving.

Quadratic Expressions

Quadratic expressions are algebraic expressions of degree 2, typically written in the form \(ax^2 + bx + c\). The term 'quadratic' comes from 'quad' implying a square, as in 'square of a term'. These expressions form the foundation for quadratic equations and show up frequently in algebra. In any quadratic expression, the key components include:

• The quadratic term: \(ax^2\), where 'a' is the coefficient.
• The linear term: \(bx\), where 'b' is the coefficient.
• The constant term: 'c', a standalone number.

In the exercise \(x^2 + 20x\), we have a quadratic expression with a quadratic term \(x^2\), and a linear term \(20x\). Notice there is no constant term, making it an incomplete square. Transforming it into a perfect square simplifies algebraic tasks like solving equations or graphing them. Quadratic expressions can often be factored or completed into perfect square trinomials for easier manipulation.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler components, called factors, that multiply together to yield the original expression. This method is highly useful for simplifying expressions and solving quadratic equations. In the case of perfect square trinomials, factoring involves expressing them as the square of a binomial. For example, consider the expression \(x^2 + 20x + 100\) that results from completing the square on \(x^2 + 20x\). This expression can be factored into \((x+10)(x+10)\) or simply \((x+10)^2\). Recognizing perfect square trinomials enables us to factor them quickly and reveals a deeper understanding of their algebraic structure. Factoring plays an important role in algebra because it simplifies solving equations and analyzing functions, thereby making it an essential skill for finding solutions and understanding the properties of quadratic expressions. When combined with completing the square, factoring enhances one’s ability to handle any complexities involving quadratic expressions efficiently.