Problem 17 - Elementary Algebra

Question

Factor. $$ x 2-13 x+12 $$

Step-by-Step Solution

Answer

The factored form is \((x-12)(x-1)\).

Step 1: Identify Coefficients

The given quadratic expression is in the form of \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = -13 \), and \( c = 12 \). We need to find two numbers that add up to \( b \) and multiply to \( c \).

Step 2: Determine Numbers for Factoring

We need two numbers whose product is 12 (the constant term, \( c \)) and whose sum is -13 (the coefficient of \( x \), \( b \) ). The numbers are -12 and -1 because \(-12 + (-1) = -13\) and \(-12 \times -1 = 12\).

Step 3: Write Factored Form

Using the numbers found in Step 2, we can write the expression \( x^2 - 13x + 12 \) as the product of two binomials: \((x - 12)(x - 1)\).

Step 4: Verify Factored Form

Expand \((x - 12)(x - 1)\) to make sure it equals the original expression. \((x - 12)(x - 1) = x^2 - x - 12x + 12 = x^2 - 13x + 12\), which matches the original expression.

Key Concepts

Coefficient Identification

Coefficient identification is the fundamental step in factoring quadratics. In any quadratic expression, like the one we’re dealing with, the structure is typically in the form \( ax^2 + bx + c \). Here, the coefficients are essentially the numbers in front of the variables or constant terms. Identifying these coefficients correctly is crucial because they determine how we will be able to factor the expression.
For the quadratic expression given as \( x^2 - 13x + 12 \), the coefficients are:

• \( a = 1 \): the coefficient of the \( x^2 \) term.
• \( b = -13 \): the coefficient of the \( x \) term.
• \( c = 12 \): the constant term.

Recognizing these coefficients helps guide us into the next steps of solving and factoring the quadratic expression correctly.

Product-Sum Method

The product-sum method is a popular technique for factoring quadratics, especially when the leading coefficient \( a = 1 \). Using the coefficients we identified, this method involves finding two numbers that:

• Multiply to give the constant term \( c \) (in this case, 12).
• Add up to give the linear coefficient \( b \) (in this case, -13).

To apply this, we need numbers whose product is 12 and sum is -13. The solution is -12 and -1, since:

• The product \(-12 \times -1 = 12\).
• The sum \(-12 + (-1) = -13\).

This pair of numbers becomes crucial in decomposing the middle term in the quadratic expression.

Binomials

Once we have determined the two numbers using the product-sum method, we can now express our quadratic as a product of two binomials. A binomial is a polynomial with two terms. In this case, we will use the numbers found (-12 and -1) to form our binomials.
The original expression \( x^2 - 13x + 12 \) can therefore be rewritten as \((x - 12)(x - 1)\). Each binomial reflects the roots or zeros of the quadratic equation, presented in their simplest forms. The beauty of using binomials is that they clearly show the intercepts where the function crosses the x-axis, providing valuable insights into the behavior of quadratic functions.

Quadratic Expressions

Quadratic expressions are a fundamental type of polynomial expression characterized by the highest power of the variable being squared, such as \( ax^2 + bx + c \). The expression \( x^2 - 13x + 12 \) is a perfect example of a quadratic expression.
The key with these expressions is understanding the significance of each term:

• The \( x^2 \) term (where \( a = 1 \)) represents the quadratic nature of the equation, which influences the parabolic shape of its graph.
• The \( x \) term (with \( b = -13 \)) impacts the slope and orientation of the parabola, affecting how it shifts along the x-axis.
• The constant \( c = 12 \) affects the parabola's vertical placement and y-intercept.

Mastering quadratic expressions leads to better mathematical problem-solving skills, as they form the basis for many further applications in algebra and geometry.