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Chapter 7: Problem 322

Solve. \((b+10)^{2}=0\)

Short Answer

Expert verified
b = -10

Step by step solution

01

Understand the Problem

The given equation is \( (b+10)^{2}=0 \). We need to find the value of \( b \) that makes this equation true.
02

Take the Square Root on Both Sides

To remove the square, take the square root of both sides of the equation: \( \sqrt{(b+10)^2} = \sqrt{0} \). This simplifies to \( b+10=0 \).
03

Solve for b

Isolate \( b \) by subtracting 10 from both sides: \( b+10-10 = 0-10 \). This simplifies to \( b = -10 \).
04

Verify the Solution

To ensure the solution is correct, substitute \( b = -10 \) back into the original equation: \( (-10 + 10)^{2} = 0 \). This simplifies to \( 0^{2}=0 \,\) confirming that the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square root
A square root is a value that, when multiplied by itself, gives the original number. In equations, taking the square root can help to simplify expressions. For example, in the given problem \( (b+10)^2=0 \), we take the square root of both sides to eliminate the exponentiation: \[ \sqrt{(b+10)^2}=\sqrt{0} \]. This reduces the complexity of the equation significantly and helps us solve it more easily. Understanding how and when to use this operation is crucial when working with quadratic equations.
algebra
Algebra involves manipulating symbols and numbers to solve equations. In our problem, once we took the square root, we simplified the problem to \( b+10=0 \). The next step is to isolate the variable \( b \). This is done through basic algebraic operations, such as addition, subtraction, multiplication, or division. Here, subtracting 10 from both sides of the equation isolates \( b \): \( b+10-10=0-10 \) leading to \( b= -10 \). Mastering these operations allows us to solve not just this problem, but a wide array of equations.
verification
Verification is the process of checking if the solution obtained is indeed correct. After solving for \( b \), we substitute \( b= -10 \) back into the original equation \( (b+10)^2=0 \). Doing the math, \( (-10+10)^2=0^2=0 \), confirms that our solution \( b= -10 \) is correct. Verification is a crucial step in problem-solving as it ensures that the answer satisfies the original equation. This step helps eliminate errors and increases confidence in the solution provided.

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