Problem 18 - Precalculus: Functions and Graphs

Question

Solve the equation by using the special quadratic equation on page 53. \((x+4)^{2}=31\)

Step-by-Step Solution

Answer

The solutions are \(x = \sqrt{31} - 4\) and \(x = -\sqrt{31} - 4\).

Step 1: Understand the Equation

The given equation is \((x+4)^2 = 31\). This is a quadratic equation in the form \((a + b)^2 = c\), where \((x+4)^2\) represents a perfect square.

Step 2: Apply the Square Root Property

To solve the equation \((x+4)^2 = 31\), apply the square root property by taking the square root of both sides. This gives:\[ x + 4 = \pm\sqrt{31} \]This means there are two potential solutions: \(x + 4 = \sqrt{31}\) and \(x + 4 = -\sqrt{31}\).

Step 3: Solve for x - Positive Solution

First, solve \(x + 4 = \sqrt{31}\) by isolating \(x\):\[ x = \sqrt{31} - 4 \]

Step 4: Solve for x - Negative Solution

Next, solve \(x + 4 = -\sqrt{31}\) by isolating \(x\):\[ x = -\sqrt{31} - 4 \]

Step 5: Verify the Solutions

Substitute both solutions back into the original equation to ensure they satisfy \((x+4)^2 = 31\). If both expressions simplify back to 31, the solutions are verified.

Key Concepts

Square Root Property

The square root property is a powerful tool used to solve quadratic equations that are expressed as a perfect square. In our given problem, the equation \((x+4)^2 = 31\) is written in such a perfect square form.
Applying the square root property involves taking the square root of both sides of the equation. When doing this, it's important to remember that taking the square root of a number results in both a positive and a negative value.
This means that you'll end up with two separate equations. In our example, you get \(x + 4 = \sqrt{31}\) and \(x + 4 = -\sqrt{31}\).
Solving these will give you the solutions to the original equation. The square root property is particularly useful because it simplifies the process of solving these types of quadratic equations.

Perfect Square

In mathematics, a perfect square is any number that is the square of an integer.
When dealing with quadratic equations like our example, recognizing a perfect square is key to applying methods like the square root property.
The given equation \((x+4)^2 = 31\) literally shows the squared term \((x+4)^2\), indicating it's a perfect square. Recognizing this form means we can efficiently use the square root property to solve it.
In general, being able to identify when an expression is a perfect square will save you time and help ensure accuracy when solving equations.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that make the equation true.
There are several methods to solve quadratic equations, such as factoring, using the quadratic formula, and completing the square. However, when the equation is in the form of a perfect square, as with \((x+4)^2 = 31\), the square root property is often the simplest method.
To solve the equation, we begin by applying the square root property. This gives us two equations: \(x + 4 = \sqrt{31}\) and \(x + 4 = -\sqrt{31}\).

• For the positive case: \(x + 4 = \sqrt{31}\), solving for \(x\) yields \(x = \sqrt{31} - 4\).
• For the negative case: \(x + 4 = -\sqrt{31}\), solving for \(x\) yields \(x = -\sqrt{31} - 4\).

In this way, quadratic equations can produce two possible solutions, and applying the correct method can help you find them efficiently.

Verification of Solutions

Verifying solutions is a crucial step in solving any equation to ensure accuracy.
After obtaining the potential solutions \(x = \sqrt{31} - 4\) and \(x = -\sqrt{31} - 4\), the next step is to substitute these back into the original equation \((x+4)^2 = 31\) to verify correctness.

• Substitute \(x = \sqrt{31} - 4\): If you simplify \([\sqrt{31} - 4 + 4]^2\), you should arrive back at 31, confirming this is a valid solution.
• Similarly, substitute \(x = -\sqrt{31} - 4\): Simplifying \([-\sqrt{31} - 4 + 4]^2\) should also return 31, confirming this solution as well.

Verification not only confirms that your solutions are accurate, but it also bolsters your confidence in the problem-solving process. Always double-check your results!