Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \((x+4)^{2}=49\)

Step-by-Step Solution

Answer

The solutions to the given quadratic equation are \(x = 3\) and \(x = -11\).

Step 1: Write down the given equation

The given equation is \((x+4)^{2}=49\)

Step 2: Apply the square root property

Applying the square root property of equations, we obtain two equations: \(x+4=\sqrt{49}\) and \(x+4=-\sqrt{49}\)

Step 3: Solve the resulting equations

Solving the first equation, we get \(x = \sqrt{49} - 4 = 7 - 4 = 3\), and solving the second equation, we get \(x = -\sqrt{49} - 4 = -7 - 4 = -11\).

Key Concepts

Square Root Property

When faced with a quadratic equation like \((x+4)^2 = 49\), the square root property is a powerful tool to simplify the solution process. This property states that if \(a^2 = b\), then \(a\) can be equal to \(\sqrt{b}\) or \(-\sqrt{b}\). In this problem, we have \((x+4)^2 = 49\). Applying the square root property means that:

• \(x+4 = \sqrt{49}\)
• \(x+4 = -\sqrt{49}\)

Remember that whenever you apply the square root, you must consider both the positive and negative roots. This is crucial in ensuring that we find all possible solutions to the quadratic equation. For \(\sqrt{49}\), the value is 7 because 7 multiplied by 7 is 49. This gives us two simpler equations to solve, one where \(x+4 = 7\) and another where \(x+4 = -7\).

Solving Equations

Once we have applied the square root property, the next step is to solve the simpler equations that result from it. Each of these equations can be solved separately to find the values of \(x\). First, consider the equation \(x+4 = 7\). To solve for \(x\), subtract 4 from both sides:\[x + 4 - 4 = 7 - 4\]This simplifies to:\[x = 3\]Similarly, for the equation \(x+4 = -7\), subtract 4 from both sides:\[x + 4 - 4 = -7 - 4\]Which simplifies to:\[x = -11\]By using these simple steps of subtracting the constant on both sides, we arrive at the solutions \(x = 3\) and \(x = -11\). It's important to check your solutions by substituting them back into the original equation to make sure they satisfy it.

Algebraic Manipulation

Algebraic manipulation is a fundamental part of solving equations, particularly when working with quadratic equations. It refers to the process of rearranging equations to isolate the variable you want to solve. In our example, after applying the square root property, we engage in algebraic manipulation to separate the variable \(x\) from other numbers in the equation. The basic operations used include:

• Addition and Subtraction: To move numbers across the equation.
• Square roots: Applied using the square root property to handle squares.

Let's focus on the step where we manipulate the equation \(x+4 = 7\). By subtracting 4 from both sides, we isolate \(x\), achieving \(x = 3\).
Similarly, in \(x+4 = -7\), we subtract 4 to find \(x = -11\).
This process of manipulating terms simplifies what appears to be complex equations into manageable components, allowing you to solve for unknowns efficiently. Make sure to always perform each arithmetic operation equally to both sides of the equation to maintain the balance and correctness of the solution.