Problem 46 - Algebra 2

Question

Solve each equation by factoring, by taking square roots, or by graphing. If necessary, round your answer to the nearest hundredth. $$ x^{2}+4 x=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = -4\).

Step 1: Understand the Equation

Identify the type of equation given. It is a quadratic equation in standard form, where the quadratic term is \(x^2\), the linear term is \(4x\), and there is no constant term.

Step 2: Factor the Quadratic Equation

Factor the left side of the equation to solve for \(x\). Factor out the common factor, which is \(x\): \(x(x+4)=0\).

Step 3: Apply the Zero Product Property

Use the zero product property, which states that if a product of factors is zero, at least one of the factors must be zero. Set both factors equal to zero: \(x = 0\) and \(x + 4 = 0\).

Step 4: Solve for x

Solve each equation from the previous step: \(x = 0\) does not need any further solving, and \(x + 4 = 0\) gives \(x = -4\). Thus, we have two solutions for \(x\).

Step 5: Write the final solution

The solutions to the quadratic equation by factoring are \(x = 0\) and \(x = -4\).

Key Concepts

Factoring Quadratics

When we work with quadratic equations like the one given, which can be written in the form \( ax^2 + bx + c = 0 \), our goal is to find the values of \( x \) for which this equation holds true. Factoring plays a crucial role in this process. Factoring a quadratic is essentially breaking it down into simpler components—more specifically, into a product of binomials or other polynomials. The initial step in our example requires us to recognize that the equation \( x^2 + 4x = 0 \) is a quadratic with a missing \( c \) term (no constant), which simplifies our task.

The trick is to find two numbers that multiply to give the product of the \( a \) coefficient (1 in this case, as it's the coefficient of \( x^2 \)) and the \( c \) term, while also adding up to the coefficient of the \( x \) term, which is 4. But since our \( c \) is zero, we only need to factor out the common \( x \), resulting in \( x(x + 4) = 0 \). This gives us a clear path to apply the zero product property and find the solutions.

Zero Product Property

The zero product property is a fundamental principle that states if you have a product of factors equal to zero, then at least one of the factors must be zero. It's the equivalent of saying if you have two numbers whose multiplication result is zero, then one or both of those numbers has to be zero. We apply this property in our example by setting each factor in the equation \( x(x + 4) = 0 \) equal to zero.

This gives us two smaller equations: \( x = 0 \) and \( x + 4 = 0 \). It simplifies the complex problem of solving a quadratic equation into solving simpler linear equations. In essence, using this property enables us to turn the daunting task of finding the roots of a quadratic equation into something much more manageable.

Quadratic Solutions

Solving a quadratic equation means finding all possible values of \( x \) that make the equation true. There can be up to two real solutions for standard quadratic equations, as they can cross the x-axis at most twice. This is symbolized graphically where the curve of the quadratic equation meets the horizontal axis.

Following the process outlined in the exercise, after factoring the quadratic and applying the zero product rule, we get two separate solutions: \( x = 0 \) and \( x = -4 \). These are the points where the graph of the quadratic equation \( x^2 + 4x = 0 \) would intersect the x-axis. Such solutions are where the quadratic reaches the height of zero, hence answering the initial problem.