Problem 5 - Algebra 1

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 that should be added is 25.

Step 1: Understand the problem.

We are given a polynomial \(x^{2}-10 x\). We want to convert this polynomial into a perfect square trinomial. A perfect square trinomial is of the form \(a^{2}+2ab+b^{2}\) or \(a^{2}-2ab+b^{2}\). Here, a = x. So let’s find the value of b.

Step 2: Formula to find 'b' of perfect square trinomial.

The perfect square trinomial formula when a = x is \(x^{2}+2bx+b^{2}\) for positive trinomial and \(x^{2}-2bx+b^{2}\) for negative trinomial. Given \(x^{2}-10 x\), this resembles the structure of the negative perfect square trinomial but it's missing \(b^{2}\).

Step 3: Applying the formula.

We know that in the given expression, -10x is equal to -2bx (where x is 'a' and 'b' is what we need to find). Hence, we equate -2bx to -10x to get the value of b. We get \(b = \frac{-10x}{-2x} = 5\). Hence, the missing term is \(b^{2} = 5^{2} = 25\).

Step 4: Conclusion.

The term that should be added to the expression \(x^{2} - 10x\) to create a perfect square trinomial is \(25\).

Key Concepts

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables. Each term is composed of a variable raised to a power, known as the degree, and a coefficient, which is a constant. In simpler terms, a polynomial can be thought of as a combination of numbers and variables using operations of addition, subtraction, and multiplication. For example, in the expression \(x^2 - 10x\), \(x^2\) and \(-10x\) are both terms that make up the polynomial.

• Terms: These are the separate elements of a polynomial; in \(x^2 - 10x\), \(x^2\) and \(-10x\) are terms.
• Coefficients: The numerical multiplier of the variable; in \(-10x\), \(-10\) is the coefficient.
• Degree: The highest power of the variable in the polynomial; for \(x^2 - 10x\), the degree is 2.

Polynomials form the building blocks for many areas in algebra and are foundational to understanding more complex algebraic expressions.

Completing the Square

Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial. This method is particularly useful for solving quadratic equations and can also help in integrating more complex algebraic expressions. The goal is to transform a quadratic polynomial into a binomial squared format.
To apply this technique, here's what you need to do for an expression like \(x^2 - 10x\):

• Identify the coefficient of \(x\), which is -10 in this case.
• Divide the coefficient by 2 to find \(b\). So, \(b = \frac{-10}{2} = -5\).
• Square \(b\) to find the term to add, which is \(b^2 = (-5)^2 = 25\).

When you add \(25\) to \(x^2 - 10x\), it transforms into \((x-5)^2\), thus completing the square. This process helps create a perfect square trinomial which is easier to factor or solve in equations.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, meaning the highest exponent of the variable is 2. A typical quadratic expression takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) is non-zero.
Quadratic expressions are fundamental in various areas of mathematics, from geometry to physics, due to their parabolic characteristics in graphs.
For the expression \(x^2 - 10x\), note the following:

• It's missing a constant term \(c\), which is why we aim to "complete the square" to balance it out as a perfect square trinomial.
• The process turns this into a simpler expression \((x-5)^2\), making it easier to understand and apply in solving equations.

Working with quadratic expressions involves techniques such as factoring, using the quadratic formula, or completing the square, each serving purposes in finding solutions to equations and analyzing the properties of quadratic functions.