Problem 64 - Elementary and Intermediate Algebra

Question

Factor $$ y^{2}+7 y+6 $$

Step-by-Step Solution

Answer

The expression factors to \((y + 1)(y + 6)\).

Step 1: Identify the Quadratic Expression

The given quadratic expression is \(y^2 + 7y + 6\). This expression is in the standard form of a quadratic equation \(ax^2 + bx + c\) with \(a = 1\), \(b = 7\), and \(c = 6\).

Step 2: Find Factors of the Constant Term

We need two numbers that multiply to the constant term \(c = 6\) and add up to the coefficient of the linear term \(b = 7\). The pairs of factors of 6 are \((1, 6)\) and \((2, 3)\).

Step 3: Choose the Correct Pair of Factors

From the factor pairs, \(1 + 6 = 7\), so \((1, 6)\) is the correct pair. These numbers satisfy both the multiplication and addition requirements: they multiply to 6 and add to 7.

Step 4: Write the Factored Form

Using the pair \((1, 6)\), rewrite \(y^2 + 7y + 6\) as \((y + 1)(y + 6)\). This is because \((y + 1)(y + 6) = y^2 + 6y + y + 6 = y^2 + 7y + 6\).

Key Concepts

Quadratic Equation

A Quadratic Equation is a key element in algebra that allows for the calculation and solving of problems involving areas, projectile motion, and more. By definition, a quadratic equation is a polynomial equation of degree two, usually taking the form of \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The variable is often referred to as \( x \), although in some cases, other variables like \( y \) can be used, as in the quadratic expression \( y^2 + 7y + 6 \). Quadratic equations are characterized by the highest degree term, which is squared. They can have real or complex roots and may be solved using various methods like factoring, the quadratic formula, or graphing. In the context of factoring, we aim to express the quadratic equation as a product of two binomials when possible. This strategy is often applied to equations with integer solutions, allowing for a simple factorization approach.

Standard Form

The Standard Form of a quadratic equation is \( ax^2 + bx + c = 0 \). This format is particularly useful because it provides a straightforward way to identify the different components of the equation and determine the values of \( a \), \( b \), and \( c \). For example, in the quadratic expression \( y^2 + 7y + 6 \), we identify:

• \( a = 1 \)
• \( b = 7 \)
• \( c = 6 \)

These coefficients are crucial when applying techniques like factoring or the quadratic formula. Knowing the standard form allows us to find the factors of the constant term and use them to break down the quadratic into simpler expressions or terms. This method can help simplify solving the equation or finding its roots. Additionally, when solving practical problems, confirming the equation is in standard form ensures all necessary components are accurately assessed.

Factored Form

Factored Form is the expression of a polynomial equation as a product of its factors. The factored form of a quadratic provides insight into the roots or solutions of the equation. For a quadratic equation like \( y^2 + 7y + 6 \), the goal is to express it as two binomial expressions, for example, \((y + p)(y + q)\), where \(p\) and \(q\) are numbers that satisfy specific conditions.To factor the quadratic \( y^2 + 7y + 6 \), one needs to identify numbers that multiply to the constant term \( c = 6 \) while also adding up to the linear coefficient \( b = 7 \). The pair \((1, 6)\) satisfies these conditions since:

• \( 1 \times 6 = 6 \) (multiplicative condition)
• \( 1 + 6 = 7 \) (additive condition)

Therefore, the factored form of the quadratic is \((y + 1)(y + 6)\). By verifying the multiplication of these factors, we ensure the original quadratic expression is correctly represented, confirming the roots of the equation are \( y = -1 \) and \( y = -6 \) when each factor is set to zero. This provides a simple yet powerful way to analyze and solve quadratic equations.