Problem 28 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$x^{2}+6 x+9=0$$

Step-by-Step Solution

Answer

Solving the equation after factoring gives the root as \(x = -3\).

Step 1: Identify a, b, and c

From the given equation \(x^2 + 6x + 9 = 0\), the \(a\), \(b\), and \(c\) values can be identified as \(a = 1\), \(b = 6\), and \(c = 9\).

Step 2: Factoring the quadratic equation

Next, the equation needs to be factored. The general form of a squared binomial is \( (x+p)^2 = x^2 + 2px + p^2 \). Here, we can see that the given equation is a perfect squared binomial with \( p = 3 \), as \( (x + 3)^2 = x^2 + 6x + 9 \). So, we have factored the expression as \( (x + 3)^2 = 0 \).

Step 3: Solve for x

Finally, solve the equation for \(x\) by setting each factor equal to zero. Thus, \(x + 3 = 0\). Solving for \(x\) will yield \(x = -3\).

Key Concepts

Quadratic Equations

Quadratic equations are foundational in algebra, involving a polynomial of degree two. Typically, these are written in the standard form as:\[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The term \( ax^2 \) is known as the quadratic term, while \( bx \) is the linear term, and \( c \) the constant term.
These equations can have either real or complex solutions, and the solutions are often where the parabola represented by the equation intersects the x-axis. Understanding these intersections helps in visualizing and solving quadratic equations. Solving quadratics can be approached using various methods, including factoring, completing the square, or applying the quadratic formula. However, when a quadratic equation is factorable, the factoring method is one of the simplest and quickest ways to find its solutions.

Factoring Method

Factoring is a method used to simplify expressions or solve equations by transforming a quadratic into a product of binomials. In the context of quadratic equations, factoring involves expressing the equation as a product of its linear components.
In our example, the equation \( x^2 + 6x + 9 = 0 \) is factored by recognizing it as a perfect square trinomial:

• We start by identifying common factors. In this case, the trinomial can be rearranged as a square of a binomial.
• The equation can be rewritten as \( (x + 3)^2 = 0 \), indicating that \( x + 3 \) is a repeated factor.
• The factor \( x + 3 \) repeated twice reflects the squared nature of the trinomial.

One of the main advantages of this method is its simplicity, particularly when the quadratic is a perfect square or easily factorable using integers. Once factored, the next step involves solving for the variable, which leads us directly to the solutions of the quadratic equation.

Solving Equations

To solve a quadratic equation after factoring, we rely on the Zero Product Property, which states if a product of two factors equals zero, at least one of the factors must be zero. By applying this property to our factored equation \( (x + 3)^2 = 0 \), we can conclude:

• Set each factor equal to zero: \( x + 3 = 0 \).
• Solve for \( x \). Here, \( x = -3 \).

This process reveals that \( x = -3 \) is the root of the equation. Because the factor \( x + 3 \) is repeated, \( x = -3 \) has a multiplicity of 2, meaning it is a double root.
After solving, it's always beneficial to verify by substituting back into the original equation or using a graphing tool to ensure that these solutions correspond to the x-intercepts of the parabola. This step verifies the accuracy of the solution and reinforces the connection between algebraic and graphical representations.