Problem 54 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ (x+4)^{2}=0 $$

Step-by-Step Solution

Answer

The solution to the equation \( (x+4)^{2}=0 \) is \( x = -4 \)

Step 1: Expand the Brackets

Let's first start by expanding the brackets: \n\n\((x+4)^{2}=0\) is equivalent to \(x^{2} + 2*4*x + 4^{2} = 0\), which simplifies to \(x^{2} + 8x + 16 = 0\).

Step 2: Set the Equation Equal to Zero

To solve for 'x', the equation must be set equal to zero. As our equation is already written in this format, we can continue to the next step.

Step 3: Solve the Equation

Now, as we are trying to solve for 'x' when the equation equals zero, look for a value of 'x' that would make the equation true. Since both terms on the left side of the equation are positive, and the equation is equal to zero, this is only possible if both terms on the left side are themselves zero. Hence, \(x + 4 = 0\). Solving this gives 'x' equal to -4.

Key Concepts

Factoring Quadratics

When handling quadratic equations, one of the most powerful techniques is factoring quadratics. This involves expressing the quadratic in a simpler multiplicative form. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Factoring means rewriting this expression as a product of binomials, such as \((px + q)(rx + s) = 0\). Doing so breaks down the quadratic into simpler parts that can easily be solved for the variable \(x\).

In the given exercise, we started with \((x+4)^{2} = 0\). This is a special case because the quadratic is already a perfect square trinomial. When expanded, it's expressed as \(x^{2} + 8x + 16 = 0\). However, we can factor it back into its original binomial form: \((x+4)(x+4) = 0\). Factoring helps identify the roots directly by looking for values of \(x\) that satisfy each binomial part of the equation.

Understanding and mastering the ability to factor quadratics simplifies solving many algebraic problems, making it an essential skill in algebra.

Zero Product Property

The zero product property is a crucial concept in algebra. It states that if the product of two numbers or expressions equals zero, then at least one of the factors must be zero. This property simplifies the task of solving for variables when dealing with equations involving products.

In our exercise, once we factor the quadratic equation \((x+4)(x+4) = 0\), the zero product property allows us to set each factor to zero individually. Because for any product to equal zero, at least one of the multiplicative components must indeed be zero:

• \(x + 4 = 0\)
• \(x + 4 = 0\)

This leads us directly to the solution \(x = -4\). Recognizing and applying the zero product property streamlines the solving process and enables quick and efficient determination of possible solutions.

Solving for x

Solving for \(x\) involves finding the values of \(x\) that satisfy the given equation. It's about isolating \(x\) on one side to determine its value. In a quadratic context, like \((x+4)^{2} = 0\), once you've factored the equation and applied the zero product property, solving for \(x\) follows naturally.

For the equation in our exercise, we determined that the factored form is \((x+4)(x+4) = 0\). Setting each factor equal to zero gives us:

• \(x + 4 = 0\)

Solving this simple linear equation requires straightforward algebraic manipulation: subtracting 4 from both sides results in \(x = -4\).

With this solution, we found that \(x = -4\) satisfies the original equation \((x+4)^{2} = 0\). This comprehensive approach ensures that we gather all valid solutions for \(x\) within the equation and strengthens your skill in tackling quadratic equations.