Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ x^{2}+10 x+25=0 $$

Step-by-Step Solution

Answer

The solution to the equation is x = -5.

Step 1: Recognize the trinomial as a perfect square

The trinomial can be seen as a perfect square because it follows the structure of \(a^{2x}+2abx+b^{2}\) with a = 1, b = 5 and x = x. So, \(x^{2}+10x+25\) can be written as \((x+5)^{2}\).

Step 2: Apply the Zero Product Property

Set the factored form of the equation equal to zero. The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. In applying this property, the equation becomes \((x+5)^{2} = 0\).

Step 3: Solve for x

Lastly, we solve for x by taking the square root on both sides of the equation, and subsequently subtracting 5 from both sides of the result. This gives x = -5.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that can be rewritten as the square of a binomial. This is because it follows a recognizable pattern: \( a^2 + 2abx + b^2 \), where \( a \) and \( b \) are constants. In the quadratic equation \( x^2 + 10x + 25 = 0 \), we can identify it as a perfect square trinomial by noticing that:

• The first term \( x^2 \) is a square of \( x \).
• The last term \( 25 \) is a square of \( 5 \).
• The middle term \( 10x \) is twice the product of \( x \) and \( 5 \).

Recognizing this form allows us to express the quadratic as \( (x + 5)^2 = 0 \). This makes it much easier to solve because we can simply take the square root on both sides. Identifying perfect square trinomials is a key step in simplifying and solving quadratic equations efficiently.

Factoring

Factoring is the process of breaking down a complex expression into simpler parts or factors that, when multiplied together, give the original expression. In the context of quadratic equations, factoring involves rewriting the equation as a product of simpler binomials. For instance, in the equation \( x^2 + 10x + 25 = 0 \), we rewrite it as a factored expression \( (x+5)^2 = 0 \). This involves recognizing patterns such as perfect square trinomials, as seen in this example.

• Factoring simplifies the process of solving equations.
• It allows for the application of other mathematical properties, such as the Zero Product Property, effectively.

Understanding how to factor quadratics is fundamental because it transforms a potentially complex task into a simple problem where we can directly find the solutions by solving for each factor separately.

Zero Product Property

The Zero Product Property is a crucial principle in algebra that helps solve equations set to zero. It states that if a product of two or more factors equals zero, at least one of the factors must be zero. This principle is highly useful in solving equations because it allows us to split the equation into simpler parts. In our example, once the quadratic \( x^2 + 10x + 25 = 0 \) is factored into \( (x+5)^2 = 0 \), the Zero Product Property tells us that \( x+5=0 \).

• This means only one solution to consider: \( x = -5 \).

By focusing on individual factors equalling zero, solving for the variable becomes direct and swift. Understanding and applying the Zero Product Property can significantly streamline solving quadratic equations, turning a complex problem into a straightforward calculation.