Problem 59 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \((x+2)^{2}=4\)

Step-by-Step Solution

Answer

The real solutions are \(x = 0\) and \(x = -4\).

Step 1: Understand the Equation

The given equation is \((x+2)^{2}=4\). We need to find the values of \(x\) that satisfy this equation. The left side of the equation is a square of the binomial \(x+2\). The right side of the equation is 4, which is a perfect square since \(4 = 2^2\).

Step 2: Set the Squared Expression Equal to its Square Roots

To solve the equation \((x+2)^2 = 4\), we take the square root of both sides. This produces two possible equations: \(x+2 = 2\) and \(x+2 = -2\), because the square root of a number can be either positive or negative.

Step 3: Solve the First Equation

Solve for \(x\) in the equation \(x+2 = 2\). Subtract 2 from both sides to isolate \(x\): \(x = 2 - 2\), which simplifies to \(x = 0\).

Step 4: Solve the Second Equation

Solve for \(x\) in the equation \(x+2 = -2\). Subtract 2 from both sides to isolate \(x\): \(x = -2 - 2\), which simplifies to \(x = -4\).

Step 5: Verify the Solutions

Substitute \(x = 0\) and \(x = -4\) back into the original equation to ensure they satisfy it. \((0+2)^2 = 4\) confirms \(x = 0\) is a solution. \((-4+2)^2 = 4\) confirms \(x = -4\) is a solution.

Key Concepts

Understanding Binomials

A binomial is a simple yet crucial concept in algebra, helping us tackle complex equations. In algebra, a binomial is an expression composed of two terms connected by a plus or minus sign.
For example, in the equation \((x+2)^2 = 4\), the section inside the parentheses, \(x+2\), forms a binomial. Binomials often appear in formulas needing calculations such as expansions or factorizations, where understanding their structure is vital for solving quadratic equations.

• A binomial is "bi-" meaning two, and "-nomial" means terms. So, a binomial has two terms.
• Binomials frequently show up in the form of \((a+b)^2\), leading to expressions that require squaring these terms.

Understanding binomials helps in approaching equations with clarity, allowing for steps like expansion and simplifications. This foundation is essential when dealing with quadratic equations like the one in our example.

Exploring Square Roots

In mathematics, taking the square root of a number means finding another number that, when multiplied with itself, gives the original number.
The significant concept about square roots is that they have two values: positive and negative.
In the example \((x+2)^2=4\), both \(+2\) and \(-2\) are squares of 4, which means:\(\sqrt{4} = 2\) and \(\sqrt{4} = -2\).

• Square roots undo the operation of squaring a number.
• They're essential for solving quadratic equations, especially when dealing with equations of the form \((a+b)^2=c\).

Understanding this concept is necessary for correctly solving quadratic equations by simplifying them and identifying potential solutions.
By recognizing both positive and negative roots, we can appropriately set up and solve down the resulting equations.

Finding Real Solutions

Real solutions refer to the values of a variable that satisfy an equation when working with real numbers.
Quadratic equations, like \((x+2)^2=4\), often have real solutions that can be found by taking the square roots of both sides of the equation.

• The real solutions of an equation are the values of \(x\) that make the equation true.
• Checking solutions found by substituting them back into the original equation ensures that they are correct.

In our example, solving \((x+2)^2=4\) led to potential solutions \(x=0\) and \(x=-4\). These results confirm values that equal the sides of the equation when substituted back, demonstrating their validity as real solutions.
This process emphasizes the critical nature of verifying solutions once they are calculated.