Problem 30 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ (b+7)^{2}=0 $$

Step-by-Step Solution

Answer

Answer: The value of b in the given equation is -7.

Step 1: Identify the given equation

The given equation is: $$(b+7)^2 = 0$$

Step 2: Apply square root property

To isolate the variable "b", we can use the square root property. Taking the square root of both sides, we get: $$\sqrt{(b+7)^2} = \sqrt{0}$$

Step 3: Simplify the equation

Now, simplify the equation further: $$b+7 = 0$$

Step 4: Solve for the variable "b"

Subtract 7 from both sides of the equation to solve for "b": $$b = -7$$ The solution for the given equation is \(b = -7\).

Key Concepts

Square Root Property

The square root property is a vital technique in solving quadratic equations, especially when the equation involves a perfect square term. When an equation is given in the form \((x+k)^2 = c\), you can apply the square root property to both sides. This allows us to "undo" the square and bring the equation into a simpler form.
By applying the square root property, we can state:

• Take the square root of both sides of the equation to cancel out the square on the left side.
• Remember that the square root of zero is zero and a perfect square leads to two identical results, thus the simplification can begin.

This method helps in directly reaching the expression without the square, aiding in isolating the variable further.

Isolate Variable

Isolating the variable is a critical step in solving any algebraic equation. The main goal here is to get the variable on one side of the equation, ideally by itself. With our example of solving \((b+7)^{2} =0\), after applying the square root property, we arrive at the simplified form:
\[b + 7 = 0\]
To isolate the variable \(b\), we'll perform algebraic operations to remove any constants or coefficients attached to it:

• Subtract the constant 7 from both sides of the equation.

This results in the variable \(b\) being isolated and gives us the solution \(b = -7\). Such manipulation is crucial to easily solve equations.

Simplification in Algebra

Simplification in algebra involves reducing equations to their simplest form to make them easier to understand and solve. It's about clarity and making an equation tractable. While simplifying an equation like \(\sqrt{(b+7)^2} = \sqrt{0}\), we execute the following:

• Recognize that taking the square root of a number and squaring it are inverse operations that cancel each other out.
• Thus, the equation simplifies directly to \(b + 7 = 0\).
• Simplification allows you to focus on solving for the variable instead of getting bogged down in complexities.

Simplifying early in the solving process saves time and reduces the chance of errors.