Problem 63 - Elementary and Intermediate Algebra

Question

Factor $$ x^{2}-5 x+6 $$

Step-by-Step Solution

Answer

The factorization of \(x^2 - 5x + 6\) is \((x - 2)(x - 3)\).

Step 1: Identify Coefficients

In the quadratic expression given, \(x^2 - 5x + 6\), observe that the coefficients are as follows: the coefficient of \(x^2\) is 1, the coefficient of \(x\) is -5, and the constant term is 6.

Step 2: Find Two Numbers

Find two numbers that multiply to the constant term (6) and add up to the coefficient of \(x\) (-5). These numbers are -2 and -3, since \((-2) \times (-3) = 6\) and \((-2) + (-3) = -5\).

Step 3: Re-write the Expression

Use the two numbers, -2 and -3, to split the middle term of the expression. Thus, we re-write \(x^2 - 5x + 6\) as \(x^2 - 2x - 3x + 6\).

Step 4: Factor by Grouping

Group the terms into two pairs:\((x^2 - 2x) + (-3x + 6)\). Factor each group separately. In the first group, factor out \(x\): \(x(x - 2)\). In the second group, factor out -3: \(-3(x - 2)\).

Step 5: Combine the Groups

Observe that \(x - 2\) is a common factor in both groups. Factor \(x - 2\) out from the entire expression: \((x - 2)(x - 3)\).

Step 6: Confirm the Factorization

Verify the factorization by expanding \((x - 2)(x - 3)\) to ensure it matches the original expression. Multiply: \(x(x - 3) - 2(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6\), which confirms that the factorization is correct.

Key Concepts

Quadratic Expressions

Quadratic expressions are mathematical expressions of the form \(ax^2 + bx + c\), where "a," "b," and "c" are constants, and "x" is a variable. These expressions are called quadratics because they involve the square of "x".
Quadratic expressions can take various forms, and one of their key characteristics is their symmetry. They often correspond to parabolic graphs that open upwards if "a" is positive, and downwards if "a" is negative.
Understanding these expressions is crucial because they appear frequently in diverse areas of mathematics, such as in solving equations and in graphical analysis. When working with quadratics, it's important to recognize the role of each term:

• The term \(ax^2\) is the quadratic term; it determines the "width" and "direction" of the parabola.
• The "b" term (\(bx\)) affects the position of the vertex in the horizontal direction.
• The constant "c" shifts the parabola vertically.

Grasping the function and interplay of these terms can greatly enhance your ability to factor and solve quadratic expressions.

Factor by Grouping

Factor by grouping is an effective method for factoring certain quadratic expressions, particularly when the middle term of the quadratic can be split conveniently. This technique often simplifies the process of factorization into more manageable parts.
Here's how you can factor a quadratic expression by grouping:

1. First, identify two numbers that multiply to give you the product of the quadratic term's coefficient and the constant term "c," while adding up to the linear term's coefficient "b".
2. Use these two numbers to split the middle term into two separate terms. This effectively transforms the original quadratic into four terms.
3. Next, group these terms into two pairs.
4. Factor out the common factors from each pair.
5. If the pairs are set up correctly, you should find a common binomial factor that can be factored out, resulting in the final factored form.

This method, though quite structured, allows for flexibility and deeper understanding of the algebraic manipulation involved. It's important to practice this technique to recognize patterns and to develop proficiency in handling quadratics.

Algebraic Techniques

Algebraic techniques form the backbone of manipulating and solving quadratic expressions. They include strategies that help simplify and understand expressions better. Among these techniques, factorization holds a special place, especially for quadratics.
When dealing with quadratics, employing algebraic techniques:

• Helps simplify expressions by breaking them down into their constituent factors which can then be managed more easily.
• Allows the transformation of complex equations into simpler parts, significantly simplifying the solving process.
• Is fundamental in finding the roots of a quadratic equation, by reducing the quadratic into linear factors.

Techniques like factorization by grouping, completing the square, and using the quadratic formula are all part of this algebraic toolkit. Each technique has its own advantages, but understanding when and how to apply them is essential.
These techniques not only aid in the resolution of current problems but also build a foundation for tackling more advanced algebraic concepts in future mathematical studies. An in-depth understanding of these techniques enriches a student's ability to manipulate algebraic expressions effectively.