Problem 56 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ 3 b^{2}=48 $$

Step-by-Step Solution

Answer

Answer: The possible values for b are 4 and -4.

Step 1: Understand the given equation

This exercise gives the equation: $$ 3 b^{2}=48 $$ Our task is to solve for b.

Step 2: Isolate the b^2 term

We start by isolating the b^2 term by dividing both sides of the equation by 3: $$ \frac{3 b^{2}}{3} = \frac{48}{3} $$ which simplifies to: $$ b^2 = 16 $$

Step 3: Take the square root of both sides

To find the value(s) of b, we take the square root of both sides of the equation: $$ \sqrt{b^2} = \sqrt{16} $$

Step 4: Solve for b

As the square root of 16 has two possible solutions, positive and negative: $$ b = \pm 4 $$ So, the possible values for b are 4 and -4.

Key Concepts

Solving Equations

Solving quadratic equations is a fundamental skill in algebra that helps us find unknown values. In this exercise, the goal is to determine the value of the variable, which is represented by \( b \) in the equation \( 3b^2 = 48 \). To solve such equations, follow these steps:

• First, isolate the term with the unknown variable. This involves moving any other terms to the opposite side of the equation. In our example, \( b^2 \) is isolated by dividing both sides by 3.
• Once the term is isolated, simplify the equation. This gives us \( b^2 = 16 \).
• Finally, solve for \( b \) by taking the square root of both sides to determine the possible values of \( b \).

Solving equations often involves balancing both sides by performing the same operations on each side. This method ensures that the equality remains true, helping you to accurately find the solution.

Square Roots

Taking square roots is a common step in solving quadratic equations. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because \( 4 \times 4 = 16 \).

• When taking the square root in an equation, such as \( b^2 = 16 \), you write it as \( \sqrt{b^2} = \sqrt{16} \).
• Calculating the square root helps us undo the square of the variable, isolating \( b \) itself.
• Remember, numbers can have two square roots: one positive and one negative. These are called the principal square root and its negative.

Understanding square roots is vital because it allows you to explore both potential solutions to the equation, broadening your understanding of how numbers and their properties work together.

Positive and Negative Solutions

Quadratic equations often yield two solutions because their graphs are parabolas. In our example, \( b^2 = 16 \) comes from such a quadratic equation. By taking the square root, we find both positive and negative solutions, \( b = \pm 4 \).

• The positive solution \( b = 4 \) suggests that \( b \) can be 4, since \( 4^2 = 16 \).
• On the flip side, \( b = -4 \) indicates that \( b \) can also be -4 because \( (-4)^2 = 16 \).
• Both values are valid because squaring either number gives us 16, satisfying the original equation \( b^2 = 16 \).

The concept of positive and negative solutions reveals that equations can have more than one valid answer. This is essential for understanding how symmetry works in mathematics, particularly in the study of functions and their graphs.