Problem 44 - Algebra 2

Question

Factor each expression. $$ x^{2}+x-6 $$

Step-by-Step Solution

Answer

The factored form of the expression \(x^{2}+x-6\) is \((x-2)(x+3)\).

Step 1: Identify the Parameters

Identify the parameters from the quadratic expression. Here these are \(a = 1\), \(b = 1\), and \(c = -6\)

Step 2: Factoring the Quadratic Expression

Look for two numbers that multiply to \(-6\) (the result of \(a*c\)) and add up to \(1\) (the coefficient of \(x\)). The numbers that satisfy these conditions are \(3\) and \(-2\). Therefore, the factored form of the expression is \((x-2)(x+3)\).

Key Concepts

Understanding Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These expressions are called "quadratic" because they involve the variable \( x \) raised to the second power, or squared. In a quadratic expression like \( x^2 + x - 6 \), you're looking at how these constants influence the shape of its graph, which is a parabola. Quadratics serve as a foundation for many areas of mathematics and are vital in understanding various real-world phenomena, such as projectile motion and areas of shapes. Understanding how to manipulate these expressions can make more complex algebraic tasks much simpler.

Mastering Factoring Techniques

Factoring techniques involve rewriting a quadratic expression as a product of simpler expressions. This method is a key skill in algebra and helps solve quadratic equations efficiently. The factored form is often easier to work with because it directly reveals the roots or solutions of the corresponding quadratic equation.

In the example \( x^2 + x - 6 \), finding two numbers that multiply to \(-6\) and add to \(1\) is the crux of factoring it. These numbers, \(3\) and \(-2\), allow you to rewrite the expression as \((x + 3)(x - 2)\). This method is known as "factoring by grouping," and it simplifies the process of solving quadratic equations.

• Look for multiplication factors of \( c \) that sum to \( b \).
• Reconstruct the expression into two binomials.

Mastering this technique is essential for solving problems and simplifying expressions quickly and accurately.

Solving Quadratic Equations

Quadratic equations are solved to find the values of \( x \) that make the equation true. This involves setting the quadratic expression equal to zero and then using various methods, such as factoring, to solve it.

Once you have the factored form of a quadratic, like \((x - 2)(x + 3) = 0\), you can easily find the solutions by setting each factor equal to zero:

• \( x - 2 = 0 \) gives \( x = 2 \).
• \( x + 3 = 0 \) gives \( x = -3 \).

Therefore, the solutions to the quadratic equation are \( x = 2 \) and \( x = -3 \). Solving these equations helps in understanding the points where the parabola intersects the x-axis, providing insights into graphical representations and real-world problem-solving. Understanding these solutions is fundamental to algebra and higher-level mathematics.