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Chapter 9: Problem 18

Solve each equation. $$\frac{3}{2} b^{2}-7=41$$

Short Answer

Expert verified
b = \pm 4 \sqrt{2}

Step by step solution

01

Simplify the Equation

Add 7 to both sides of the equation to isolate the term with the variable. \[\frac{3}{2} b^{2} - 7 + 7 = 41 + 7\]This simplifies to: \[\frac{3}{2} b^{2} = 48\]
02

Isolate the Variable

Multiply both sides by the reciprocal of \(\frac{3}{2}\) to solve for \( b^2 \). \[\frac{2}{3} \cdot \frac{3}{2} b^{2} = 48 \cdot \frac{2}{3}\]This simplifies to: \[b^{2} = 32\]
03

Solve for \( b \)

Take the square root of both sides to solve for \( b \). \[b = \pm \sqrt{32}\]Simplify the square root: \[b = \pm 4 \sqrt{2}\]

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Key Concepts

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

Isolating Variables
Isolating variables is a fundamental step in solving equations. It means getting the variable you are solving for on one side of the equation, without any other terms attached. This makes it easier to find the value of the variable.
In the exercise, we needed to solve for \( b \) in the equation: \[ \frac{3}{2} b^2 - 7 = 41 \].
First, we added 7 to both sides to isolate the term with the variable: \[ \frac{3}{2} b^2 = 48 \].
We can think of isolating variables as peeling away layers to get to the core term. Common steps include:
  • Adding or subtracting numbers from both sides
  • Multiplying or dividing both sides by the same value
These steps ensure that the equation remains balanced while simplifying the path to find the variable.
Simplifying Expressions
Simplifying expressions is crucial for making equations more manageable. It involves reducing expressions to their simplest form by performing operations and combining like terms.
In the given exercise, after isolating the term with the variable, we had: \[ \frac{3}{2} b^2 = 48 \].
To simplify this, we multiplied both sides by the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \): \[ \frac{2}{3} \times \frac{3}{2} b^2 = 48 \times \frac{2}{3} \].
These steps simplified the equation to: \[ b^2 = 32 \].
When simplifying expressions, we use various techniques such as:
  • Combining like terms
  • Using distributive properties
  • Applying appropriate arithmetic operations
These help in maintaining the integrity of the equation while reducing complexity.
Square Root Properties
Square root properties help us solve equations involving square terms. When we have an equation like \( b^2 = 32 \), we use square roots to find the value of \( b \).
In the final step of the exercise, we took the square root of both sides: \[ b = \pm \sqrt{32} \].
This resulted in: \[ b = \pm 4\sqrt{2} \].
It is essential to remember:
  • Taking the square root of a number gives both positive and negative roots
  • The square root of a product can be separated into the square roots of the factors
That means \( \sqrt{32} \) can be broken down as \( 4\sqrt{2} \) since \( 32 \) is equal to \( 16 \times 2 \) and \( \sqrt{16} = 4 \). Knowing these properties allows us to simplify and solve square root equations efficiently.

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Most popular questions from this chapter

Solve. Round answers to the nearest tenth. An arrow is shot vertically upward from a platform 45 feet high at a rate of \(168 \mathrm{ft} / \mathrm{sec}\). Use the quadratic function \(h(t)=-16 t^{2}+168 t+45\) find how long it will take the arrow to reach its maximum height, and then find the maximum height.

Graph each function using transformations. $$f(x)=(x-4)^{2}-3$$

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=2 x^{2}-12 x+7$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}+4 x+3$$

Graph each function using a horizontal shift. $$f(x)=(x+2)^{2}$$
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