Problem 45 - Algebra 2

Question

Factor each expression. $$ x^{2}-7 x+10 $$

Step-by-Step Solution

Answer

The factored form is \( (x-5)(x-2) \)

Step 1: Identify the Quadratic Formula

Optically recognize the quadratic formula in standard form \( ax^{2} + bx + c \). In this problem, it is given as \( x^{2} - 7x + 10 \) where \( a=1 \), \( b=-7 \), and \( c=10 \).

Step 2: Find two numbers

Find two numbers that when multiplied together give \( c \), and when added together give \( b \). In this case, -5 and -2 multiplied together is +10 and added together is -7.

Step 3: Write in the factored form

The original equation can thus be expressed as \( (x-a) (x-b) \), replacing \( a \) and \( b \) with the two numbers that you have found from the previous step. Thus, factored form of the given quadratic equation is \( (x-5)(x-2) \).

Key Concepts

Quadratic Equations

Quadratic equations are expressions that involve unknown variables, specifically squared terms. These equations generally take the form \( ax^{2} + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. Because of the squared term, each quadratic equation can potentially have two solutions.

Solving quadratic equations often involves finding the roots of the equation. Roots are the values of the variable \( x \) that make the equation true, or "zero." One of the essential methods of finding these roots is by factoring, which involves rewriting the quadratic as a product of simpler expressions.

Understanding how to manipulate quadratic equations is a fundamental skill in algebra, serving as a foundation for more complex mathematical concepts. Mastery in handling these equations opens the door to solving diverse problems in mathematics and various applied fields.

Factoring Techniques

Factoring techniques are methods used to simplify algebraic expressions by breaking them down into smaller, more manageable components. For quadratic equations, factoring can reveal the solutions or roots of the equation.

One common technique is looking for two numbers that meet these criteria:

• They multiply to give the constant term \( c \).
• They add up to the coefficient \( b \) of the linear term \( x \).

In the example \( x^{2} - 7x + 10 \), the numbers \(-5\) and \(-2\) satisfy these conditions because:

• \((-5) \times (-2) = +10\), which is our \( c \).
• \((-5) + (-2) = -7\), which matches our \( b \).

Thus, these numbers help represent the quadratic into its factored form as \((x - 5)(x - 2)\). This method leverages patterns and strategic assumptions to arrive at a solution more efficiently.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. They provide a rich framework for analyzing and solving equations.

A quadratic equation such as \( x^{2} - 7x + 10 \) is a second-degree polynomial because the highest exponent of the variable is 2. Polynomials form the backbone of algebraic expressions, allowing us to generalize mathematical concepts like factoring and root finding.

The degree of a polynomial determines many properties, such as the number of possible roots. Understanding the foundational concepts of polynomials can assist in various aspects of mathematics, including calculus, where they appear in more complex forms.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operators, building blocks that allow us to describe a wide array of mathematical realities. They can be as simple as a single variable \( x \), or complex like a quadratic equation.

In our focus example, \( x^{2} - 7x + 10 \), we explore an algebraic expression that represents a specific problem in a streamlined way. The challenge is to manipulate this expression using operations like factoring.

Understanding algebraic expressions involves recognizing different forms and applying suitable techniques to simplify or solve them. They are essential tools for expressing complex ideas succinctly and finding precise solutions in both mathematical theory and practical applications.