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Chapter 7: Problem 24

Expand each expression. $$ -2 b(8 b-0.5) $$

Short Answer

Expert verified
-16b^2 + b

Step by step solution

01

Distribute \( -2b \)

To expand the expression, distribute \( -2b \) to each term inside the parentheses. The expression inside the parentheses is \( (8b - 0.5) \), so multiply \( -2b \) by each term inside the parentheses.
02

Multiply \( -2b \) by \( 8b \)

First, multiply \( -2b \) by \( 8b \): \[ -2b \times 8b = -16b^2 \]
03

Multiply \( -2b \) by \( -0.5 \)

Next, multiply \( -2b \) by \( -0.5 \): \[ -2b \times -0.5 = 1b = b \]
04

Combine the results

Combine the results from Step 2 and Step 3 to obtain the expanded expression: \[ -16b^2 + b \]

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Key Concepts

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

distributive property
The Distributive Property is a useful tool in algebra. It helps us expand expressions by distributing a number outside the parentheses to each term inside the parentheses.
Consider the expression: \( -2b(8b - 0.5) \). Using the Distributive Property, we multiply \( -2b \) by each term inside the parentheses separately:

  • Step 1: Multiply \( -2b \) by \( 8b \)
  • Step 2: Multiply \( -2b \) by \( -0.5 \)
After performing these steps, we combine the results to get our expanded expression.
multiplying monomials
When multiplying monomials, it's essential to multiply the coefficients (the numerical parts) and the variables separately.

For instance, in the expression \( -2b \times 8b \):

  • First, multiply the coefficients: \( -2 \times 8 = -16 \)
  • Then, multiply the variables: \( b \times b = b^2 \)
Combining these results gives us the term: \( -16b^2 \).

Next, let's multiply \( -2b \times -0.5 \):
  • Multiply the coefficients: \( -2 \times -0.5 = 1 \)
  • Attach the variable: \( 1 \times b = b \)
Combining these results gives us the term: \( b \).
combining like terms
Once we multiply the terms, we need to combine like terms to simplify the expression.
In the given example, we obtained the terms \( -16b^2 \) and \( b \).

  • Like terms are terms that have the same variable raised to the same power.
  • In our case, \( -16b^2 \) and \( b \) are not like terms since their variables are raised to different powers.
Since they are not like terms, we simply combine them to form our final expanded expression:
\( -16b^2 + b \)

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Most popular questions from this chapter

When you start the process of completing the square for an equation, you may be able to tell whether the equation has solutions without solving it. a. Express each of these using a perfect square plus a constant. Without solving, decide whether the equation has a solution, and explain your answer. $$ \begin{array}{l}{\text { i. } x^{2}+6 x+15=0} \\ {\text { ii. } x^{2}+6 x+5=0}\end{array} $$ b. State a rule for determining whether an equation of the form \((x+a)^{2}+c=0\) has solutions. Explain your rule.

Solve each equation by factoring using integers, if possible. If an equation can't be solved in this way, explain why. $$ u^{2}+5 u=36 $$

Consider the quadratic relationship \(y=x(x-1)\) a. For what values of \(x\) is \(y=0\) ? b. Is \(y\) positive or negative for \(x\) values between those you listed in Part a? c. Can \(y\) ever be equal to \(-1 ?\) Explain. d. Can \(y\) ever be equal to 1\(?\) Explain. e. Sketch a graph of this relationship. f. Challenge Use your knowledge of the quadratic formula and your graph to find the minimum value of \(y .\)

Because 7 isn't a perfect square, the expression \(4 x^{2}-7\) doesn't look like the difference of two squares. But 7 is the square of something. a. What is 7 the square of? b. How can you use your answer to Part a to factor \(4 x^{2}-7\) into a product of two binomials?

Solve each equation. $$ (x-5)(x+7)=0 $$
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