Problem 28 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ x^{2}+7 x+6=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = -6 \) and \( x = -1 \).

Step 1: Identify the Equation Type

The equation given is a quadratic equation because it has the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 7 \), and \( c = 6 \).

Step 2: Use the Factoring Method

We try to factor the quadratic expression \( x^2 + 7x + 6 \) into two binomials. Look for two numbers that multiply to 6 (the constant term) and add to 7 (the coefficient of \( x \)).

Step 3: Find the Factors

The two numbers that multiply to 6 and add to 7 are 6 and 1. Therefore, we can factor the expression as \((x + 6)(x + 1) = 0\).

Step 4: Apply the Zero Product Property

Using the zero product property, set each factor equal to zero: \( x + 6 = 0 \) and \( x + 1 = 0 \).

Step 5: Solve for x

Solve each equation: \( x + 6 = 0 \) gives \( x = -6 \) and \( x + 1 = 0 \) gives \( x = -1 \).

Step 6: State the Solution

The solutions to the equation are \( x = -6 \) and \( x = -1 \).

Key Concepts

Factoring

Factoring is a method used to simplify quadratic equations and find their solutions. In fact, it involves expressing a quadratic equation in the form of a product of two binomials. Consider the quadratic equation given:

• Identify the quadratic equation: The equation is in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 7 \), and \( c = 6 \).
• Find two numbers that multiply to \( c \) (which is 6 in this case), and add up to \( b \) (which is 7).

For this equation, the numbers are 6 and 1, since \( 6 \times 1 = 6 \) and \( 6 + 1 = 7 \). This lets you factor the expression as \( (x + 6)(x + 1) = 0 \).Factoring is essential because it simplifies the quadratic equation into linear factors, making it easier to solve. Not all quadratics can be factored easily; if factoring is challenging, other methods like the quadratic formula or completing the square may be used. But when it works, factoring is quick and straightforward!
Enhance your understanding by practicing with more examples and get comfortable with spotting the factors quickly.

Zero Product Property

The zero product property is a powerful tool in algebra that allows us to solve equations once the expression is factored. This property states that if the product of two numbers is zero, then at least one of the factors must be zero. Therefore, if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).
Let's apply this to our factorized equation:

• We factored the quadratic equation as \( (x + 6)(x + 1) = 0 \).
• To solve for \( x \), set each factor equal to zero: \( x + 6 = 0 \) and \( x + 1 = 0 \).

This straightforward step ensures that we find all possible values of \( x \) that satisfy the original equation. The zero product property makes solving quadratic equations efficient and straightforward after factoring is complete. Being familiar with this property can help you swiftly navigate through algebraic solutions involving products set to zero.

Solving Equations

Solving equations is a fundamental skill in mathematics, particularly for quadratic equations, which are common in algebra and real-world applications.
Here’s how to solve the factored equation:

• After factoring and applying the zero product property, the two simple equations are \( x + 6 = 0 \) and \( x + 1 = 0 \).
• Solving these linear equations is straightforward.
  • For \( x + 6 = 0 \), subtract 6 from both sides to find \( x = -6 \).
  • For \( x + 1 = 0 \), subtract 1 from both sides to find \( x = -1 \).

Thus, the solutions to the original quadratic equation \( x^2+7x+6=0 \) are \( x = -6 \) and \( x = -1 \).
This sequence involves identifying the type of equation, using factoring to break it down, applying the zero product property, and finally solving resulting equations. Practicing these steps with various problems will make you proficient in tackling quadratic equations across different contexts.