Problem 64 - Algebra 2

Question

What can you add to \(x^{2}+5 x\) to get a perfect square trinomial? \(\begin{array}{llll}{\text { A. } \frac{25}{4}} & {\text { B. } \frac{25}{2}} & {\text { C. } 25} & {\text { D. } 2.5 x}\end{array}\)

Step-by-Step Solution

Answer

\(\frac{25}{4}\)

Step 1: Understand what a perfect square trinomial is

A perfect square trinomial is a quadratic expression of the form \(a x^{2} + 2 a b x + b^{2}\) that can be factored into \( (a x + b)^{2} \). We want to make the expression \(x^{2}+5 x\) into a perfect square trinomial.

Step 2: Use the perfect square formula to determine the term to add

A perfect square trinomial takes the form \( (x + c)^{2} \). Expanding this gives \( x^{2} + 2 cx + c^{2}\). To make \(x^{2}+5 x\) a perfect square, it needs a constant term \(c^{2}\) where \(2 c\) is equal to the coefficient of \(x\), which is 5.

Step 3: Solve for the constant term \(c^{2}\)

Since \(2 c = 5\), divide 5 by 2 to solve for \(c\): \(\frac{5}{2}\). Now square \(c\) to get \(c^{2}\): \((\frac{5}{2})^{2} = \frac{25}{4}\).

Step 4: Conclude the correct answer

Adding \(c^{2} = \frac{25}{4}\) to \(x^{2} + 5 x\) will result in a perfect square trinomial. Therefore, the answer is \(\frac{25}{4}\), which corresponds to option A.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomials of degree two. They generally take the form \( ax^{2} + bx + c \) where \( a \), \( b \) and \( c \) are constants, and \( a \) is not equal to zero. The shape of their graph is a parabola which can either open upwards or downwards, depending on the sign of \( a \).

In the context of our problem, \( x^{2}+5x \) is a quadratic expression missing the \( c \) term to complete the square. To transform this expression into a perfect square trinomial, one must understand how quadratic expressions behave and how their terms relate to the visual representation on the graph.

Factoring Quadratics

Factoring quadratics is an essential technique in algebra. It involves breaking down a quadratic expression into a product of two binomials. For a perfect square trinomial, the factored form is especially simple, taking the structure of \( (ax + b)^{2} \) where the squared binomial indicates that the expression is the result of multiplying a binomial times itself.

Understanding this can help you quickly identify a perfect square trinomial and reverse engineer the process to complete the square in an incomplete quadratic expression. If we take our expression \( x^{2}+5x \) and wish to factor it as if it were a perfect square trinomial, it would look like \( (x + d)^{2} \) for some number \( d \) which we need to determine.

Completing the Square

Completing the square is a method used to solve quadratic equations and also to change the form of a quadratic expression to make it a perfect square trinomial. This process involves adding a specific term to the expression that allows factoring into a binomial square \( (x + c)^{2} \).

To complete the square for \( x^{2}+5x \), we look for a term \( c^{2} \) such that when added, the expression becomes \( x^{2} + 5x + c^{2} = (x + c)^{2} \). As shown in the step-by-step solution, we calculate \( c \) by taking half of the \( x \) coefficient (in this case \( 5/2 \) and then squaring it. Hence, by adding \( (5/2)^{2}= 25/4 \), we complete the square and create the perfect square trinomial \( x^{2} + 5x + 25/4 \) which factors to \( (x + 5/2)^{2} \).