Problem 23 - Elementary Algebra

Question

For the following problems, solve the equations. $$ 2 \sqrt{a+2}-2=0 $$

Step-by-Step Solution

Answer

Answer: -1

Step 1: Move the constant to the other side

To begin, let's isolate the square root term by moving the constant term to the other side. We will add 2 to both sides of the equation: $$ 2 \sqrt{a+2}-2+2=0+2 $$ This simplifies to: $$ 2 \sqrt{a+2}=2 $$

Step 2: Square both sides of the equation

Next, we need to eliminate the square root. To do this, we will square both sides of the equation. Be careful to apply the square to both terms on the left side: $$ (2 \sqrt{a+2})^2=2^2 $$ Squaring each side, we get: $$ 4(a+2)=4 $$

Step 3: Solve for 'a'

Now, let's solve for 'a'. We will first simplify the equation by distributing the 4 on the left side: $$ 4a+8=4 $$ Next, subtract 8 from both sides: $$ 4a=-4 $$ Finally, divide by 4 to find the value of 'a': $$ a=-1 $$ Therefore, the solution to the given equation is \(a=-1\).

Key Concepts

Square Roots

Square roots are a fundamental concept in algebra that represent a number which, when multiplied by itself, gives the original number under the root. In our equation, the expression within the square root is \(a+2\). Square roots are often represented using the radical symbol \(\sqrt{}\).

To solve equations involving square roots, it is important to isolate the term containing the square root first. This is because squaring both sides of the equation is the typical method used to remove or "undo" the square root. It's crucial to handle square roots carefully, as they can introduce extraneous solutions—solutions that do not satisfy the original equation.

• Check if the solution works by substituting it back into the original equation.
• Remember that every positive number has two square roots: one positive and one negative. In real solutions, we only consider the principal (positive) square root.

Using these principles, we found that after isolating and squaring the equation, the solution was confirmed as \(a = -1\).

Isolating Variables

Isolating the variable is a critical step in solving algebraic equations. It involves rearranging the equation so that the variable you need to solve for stands alone on one side of the equation. This makes it easier to determine its value.

In our problem, we began by isolating the term with the square root. This was done by moving the constant (-2) to the other side of the equation. Here's why this step is essential:

• It simplifies the equation to focus on the problematic part (in this case, the square root).
• Isolating the square root allowed us to apply further algebraic techniques, like squaring, without added complications.

Once isolated, it was simple to manipulate and eventually solve for the variable. This is a powerful tactic in algebra that applies to nearly all types of equations.

Distributive Property

The distributive property is a fundamental concept used in algebra to simplify expressions and equations. In its basic form, it states that \(a(b + c) = ab + ac\). This property is vital when we have an equation that involves multiplication over addition or subtraction.

In our solved equation, after squaring both sides, we used the distributive property to expand \(4(a+2)\) into \(4a + 8\). This step lays the groundwork for easier algebraic manipulation later on.

• Using the distributive property correctly ensures that we multiply each term within the parenthesis by the factor outside.
• This step keeps equations manageable and sets them up for simpler resolution as seen when solving for 'a'.

The distributive property helps eliminate grouping symbols and combines like terms, making complex equations significantly easier to tackle.

Algebraic Manipulation

Algebraic manipulation refers to the various techniques used to move or change parts of an equation to solve for unknowns. The tactics include using operations like addition, subtraction, multiplication, and division strategically.

In solving the given equation, several algebraic manipulations were employed:

• Adding 2 to both sides to set the equation to \(2\sqrt{a+2} = 2\).
• Squaring both sides to remove the square root resulted in expanding terms with the distributive property.
• Subtracting 8 from both sides and dividing by 4 in solving \(4a + 8 = 4\).

Each of these manipulations was an important step towards isolating and solving for the variable 'a'. This adaptability is at the heart of algebra, turning complex equations into simple, solvable forms.