Problem 60 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate. $$5(x+2)^{2}+1=16$$

Step-by-Step Solution

Answer

The solutions are \(x = -2 + \sqrt{3}\) and \(x = -2 - \sqrt{3}\).

Step 1: Simplify the Equation

First, we need to simplify the given quadratic equation. The equation is \[5(x+2)^2 + 1 = 16\]Start by isolating the squared term by subtracting 1 from both sides:\[5(x+2)^2 = 15\]

Step 2: Divide by 5

Divide both sides of the equation by 5 to solve for \((x+2)^2\):\[(x+2)^2 = 3\]

Step 3: Take the Square Root

To eliminate the square, take the square root of both sides of the equation:\[x+2 = \pm\sqrt{3}\]Remember that taking the square root introduces a positive and negative solution.

Step 4: Solve for x

Now, solve for x by subtracting 2 from both sides:\[x = -2 + \sqrt{3}\]\[x = -2 - \sqrt{3}\]These are the two possible solutions for x.

Key Concepts

Solving Quadratic Equations

To grasp quadratic equations, it's essential to understand their basic form, which is generally expressed as \(ax^2 + bx + c = 0\). A quadratic equation is a type of polynomial with a degree of 2, which means it has the variable raised to the power of 2 as its highest term. Solving these equations involves finding values of \(x\) that satisfy the equation. Typically, there are several methods to solve quadratic equations, such as factoring, completing the square, and using the quadratic formula.

In our specific case, \(5(x+2)^2 + 1 = 16\), we will explore unraveling the equation through simplification and the square root method. By applying the most suitable method, depending on the equation type and structure, we effectively find the solutions for \(x\). Whether the solutions are real or complex, our aim is always the same: to make the equation equal zero by understanding and applying mathematical principles correctly.

Square Root Method

The Square Root Method is a straightforward way to solve quadratic equations in the form \((ax+b)^2 = c\). The equation in such a setting allows us to isolate the squared expression easily. The basic idea involves eliminating the square by taking the square root of both sides.

For our example, after simplifying to \((x+2)^2 = 3\), the square root method is applied. By taking the square root of both sides, we get \(x+2 = \pm\sqrt{3}\). It is crucial to remember that this process introduces both the positive and negative roots. This step ensures that we consider all possible solutions.

Therefore, when employing the square root method, always bear in mind:

• Introduce both positive and negative solutions.
• Check your work for possible errors in calculation.
• Ensure that the equation is fully simplified before applying the method.

This method is particularly handy when the quadratic can be easily expressed as a perfect square, allowing us to efficiently progress in our solution.

Equation Simplification

Before employing any method to solve a quadratic equation, simplifying it is essential. Simplification makes the equation more manageable and ensures clarity in the steps that follow.

In the example \(5(x+2)^2 + 1 = 16\), simplification begins with isolating the squared term. First, subtract 1 from both sides to get \(5(x+2)^2 = 15\). Then, simplify further by dividing both sides by 5, resulting in \((x+2)^2 = 3\).

This step reveals the equation's underlying structure and prepares it for solving through other methods, like the square root method or factoring. Simplification involves meticulous work with arithmetic operations, keeping track of each alteration, and maintaining the balance of the equation.

Remember:

• Perform operations equally on both sides.
• Focus on isolating terms for clarity.
• Simplify iteratively and check each step.

Proper simplification lays the groundwork for efficient problem-solving, ensuring you're ready to tackle the equation with more advanced steps.

Isolating Terms

Isolating terms is a critical skill in solving algebraic equations. By isolating terms, particularly the variable of interest, you prepare the equation for the final steps in solving. This concept primarily involves rearranging equations through algebraic operations such as addition, subtraction, multiplication, and division.

In the context of \((x+2)^2 = 3\) derived from the original quadratic equation, the isolating process starts even earlier when simplifying \(5(x+2)^2 + 1 = 16\). After simplifying, isolating helps focus on the term involving \(x\), where subtraction of 2 from both sides leads to our solutions: \(x = -2 + \sqrt{3}\) and \(x = -2 - \sqrt{3}\).

Key considerations for effectively isolating terms:

• Identify which terms need to be isolated for solving.
• Use inverse operations to move terms across the equation.
• Constantly check for errors to avoid misplacement.

Correct isolation simplifies the problem significantly, ensuring that once terms are isolated, solving for \(x\) becomes straightforward. With careful manipulation, you maintain accuracy and progress smoothly through solution steps.