## Engage NY Eureka Math Algebra 1 Module 4 Lesson 12 Answer Key

### Eureka Math Algebra 1 Module 4 Lesson 12 Example Answer Key

Example 1.
Now complete the square for 2x2 + 16x + 3.
Now that students are comfortable with rewriting expressions by completing the square, we can introduce expressions with leading coefficients other than 1. Start by writing the quadratic expression below on the board or screen. Then, walk students through the process of completing the square.
2x2 + 16x + 3
Since students already know how to complete the square when the leading coefficient is 1, one way to deal with the leading coefficient is to group the x – terms and factor out the leading coefficient. Then, they can proceed exactly as they did in the previous lesson. Students should be careful to pay attention to the multiplier on the outside of the parentheses and also to the signs involved.
2(x2 + 8x ) + 3
Now complete the square of the quadratic expression in the parentheses, and offset the addition on the outside of the parentheses.
2(x2 + 8x + 42 ) + 3 – 2(42)
Make sure all agree that the two operations will reverse each other and that the new expression is equivalent to the old.
2(x + 4)2 + 3 – 32 → 2(x + 4)2 – 29
Check:
2(x + 4)2 – 29=2(x2 + 8x + 16) – 29=2x2 + 16x + 32 – 29=2x2 + 16x + 3
Yes, this matches our original expression.

Example 2.
UNIT PRICE (PRICE PER UNIT): The price per item a business sets to sell its product, which is sometimes represented as a linear expression.
QUANTITY: The number of items sold, sometimes represented as a linear expression.
REVENUE: The total income based on sales (but without considering the cost of doing business).
UNIT COST (COST PER UNIT) OR PRODUCTION COST: The cost of producing one item, sometimes represented as a linear expression.
PROFIT: The amount of money a business makes on the sale of its product. Profit is determined by taking the total revenue (the quantity sold multiplied by the price per unit) and subtracting the total cost to produce the items (the quantity sold multiplied by the production cost per unit): Profit=Total Revenue – Total Production Costs.
We can integrate the linear relationship of selling price to quantity and the profit formula to create a quadratic equation, which we can then maximize.

The following business formulas are used in this and the remaining lessons in the module:
Total Production Costs=(cost per unit)(quantity of items sold)
Total Revenue=(price per unit)(quantity of items sold)
Profit=Total Revenue – Total Production Costs
Now solve the following problem:

### Eureka Math Algebra 1 Module 4 Lesson 12 Problem Set Answer Key

Question 1.
– 2x2 + 8x + 5
– 2(x2 – 4x + 4) + 5 + 8 → – 2(x – 2)2 + 13

Question 2.
2.5k2 – 7.5k + 1.25
2.5(k2 – 3k + 2.25) + 1.25 – 5.625 → 2.5(k – 1.5)2 – 4.375

Question 3.
$$\frac{4}{3}$$ b2 + 6b – 5
$$\frac{4}{3}$$ (b2 + $$\frac{9}{2}$$ b + $$\frac{81}{16}$$) – 5 – $$\frac{27}{4}$$ → $$\frac{4}{3}$$ (b + $$\frac{9}{4}$$)2 – $$\frac{47}{4}$$

Question 4.
1000c2 – 1250c + 695
1000(c2 – 1.25c + 0.6252 ) + 695 – 390.625 → 1000(c – 0.625)2 + 304.375

Question 5.
8n2 + 2n + 5
8(n2 + $$\frac{1}{4}$$ n + $$\frac{1}{64}$$) + 5 – $$\frac{1}{8}$$ → 8(n + $$\frac{1}{8}$$)2 + 4 $$\frac{7}{8}$$

### Eureka Math Algebra 1 Module 4 Lesson 12 Exit Ticket Answer Key

Question 1.
Complete the square: ax2 + x + 3.
a(x2 + $$\frac{1}{a}$$ x) + 3 → a(x + $$\frac{1}{2a}$$)2 + 3 – $$\frac{1}{4a}$$
The text passage contains multiple examples and problems related to completing the square for quadratic expressions, with solutions provided for each.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
## Engage NY Eureka Math Algebra 1 Module 4 Lesson 12 Answer Key

### Eureka Math Algebra 1 Module 4 Lesson 12 Example Answer Key

Example 1.
Now complete the square for 2x2 + 16x + 3.
Now that students are comfortable with rewriting expressions by completing the square, we can introduce expressions with leading coefficients other than 1. Start by writing the quadratic expression below on the board or screen. Then, walk students through the process of completing the square.
2x2 + 16x + 3
Since students already know how to complete the square when the leading coefficient is 1, one way to deal with the leading coefficient is to group the x – terms and factor out the leading coefficient. Then, they can proceed exactly as they did in the previous lesson. Students should be careful to pay attention to the multiplier on the outside of the parentheses and also to the signs involved.
2(x2 + 8x ) + 3
Now complete the square of the quadratic expression in the parentheses, and offset the addition on the outside of the parentheses.
2(x2 + 8x + 42 ) + 3 – 2(42)
Make sure all agree that the two operations will reverse each other and that the new expression is equivalent to the old.
2(x + 4)2 + 3 – 32 → 2(x + 4)2 – 29
Check:
2(x + 4)2 – 29=2(x2 + 8x + 16) – 29=2x2 + 16x + 32 – 29=2x2 + 16x + 3
Yes, this matches our original expression.

Example 2.
UNIT PRICE (PRICE PER UNIT): The price per item a business sets to sell its product, which is sometimes represented as a linear expression.
QUANTITY: The number of items sold, sometimes represented as a linear expression.
REVENUE: The total income based on sales (but without considering the cost of doing business).
UNIT COST (COST PER UNIT) OR PRODUCTION COST: The cost of producing one item, sometimes represented as a linear expression.
PROFIT: The amount of money a business makes on the sale of its product. Profit is determined by taking the total revenue (the quantity sold multiplied by the price per unit) and subtracting the total cost to produce the items (the quantity sold multiplied by the production cost per unit): Profit=Total Revenue – Total Production Costs.
We can integrate the linear relationship of selling price to quantity and the profit formula to create a quadratic equation, which we can then maximize.

The following business formulas are used in this and the remaining lessons in the module:
Total Production Costs=(cost per unit)(quantity of items sold)
Total Revenue=(price per unit)(quantity of items sold)
Profit=Total Revenue – Total Production Costs
Now solve the following problem:

### Eureka Math Algebra 1 Module 4 Lesson 12 Problem Set Answer Key

Question 1.
– 2x2 + 8x + 5
– 2(x2 – 4x + 4) + 5 + 8 → – 2(x – 2)2 + 13

Question 2.
2.5k2 – 7.5k + 1.25
2.5(k2 – 3k + 2.25) + 1.25 – 5.625 → 2.5(k – 1.5)2 – 4.375

Question 3.
$$\frac{4}{3}$$ b2 + 6b – 5
$$\frac{4}{3}$$ (b2 + $$\frac{9}{2}$$ b + $$\frac{81}{16}$$) – 5 – $$\frac{27}{4}$$ → $$\frac{4}{3}$$ (b + $$\frac{9}{4}$$)2 – $$\frac{47}{4}$$

Question 4.
1000c2 – 1250c + 695
1000(c2 – 1.25c + 0.6252 ) + 695 – 390.625 → 1000(c – 0.625)2 + 304.375

Question 5.
8n2 + 2n + 5
8(n2 + $$\frac{1}{4}$$ n + $$\frac{1}{64}$$) + 5 – $$\frac{1}{8}$$ → 8(n + $$\frac{1}{8}$$)2 + 4 $$\frac{7}{8}$$

### Eureka Math Algebra 1 Module 4 Lesson 12 Exit Ticket Answer Key

Question 1.
Complete the square: ax2 + x + 3.
a(x2 + $$\frac{1}{a}$$ x) + 3 → a(x + $$\frac{1}{2a}$$)2 + 3 – $$\frac{1}{4a}$$
user
The text passage contains multiple examples and problems related to completing the square for quadratic expressions, with solutions provided for each.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS