## Math 8 Chapter 1 Lesson 3: Memorable Equivalence Constants (continued)

## 1. Theoretical Summary

### 1.1. Square of a sum

\({\left( {A + B} \right)^2} = {A^2} + 2AB + {B^2}\)

### 1.2. Square of a difference

\({\left( {A – B} \right)^2} = {A^2} – 2AB + {B^2}\)

### 1.3. Difference of two squares

\({A^2} – {B^2} = \left( {A – B} \right)\left( {A + B} \right)\)

We can easily prove these eigenvalues by multiplying the polynomial by the polynomial we learned in the previous lesson.

## 2. Illustrated exercise

**Question 1. **Mental arithmetic:

a.\({99^2}\)

b.\({102^2}\)

**Solution guide**

**Question a:**

\(\begin{array}{l} {99^2} = {(100 – 1)^2}\\ = {100^2} – 2,100 + 1\\ = 10000 – 200 + 1 = 9801 \end{ array}\)

**Sentence b:**

\(\begin{array}{l} {102^2} = {(100 + 2)^2}\\ = {100^2} + 2.2.100 + {2^2}\\ = 10000 + 400 + 4 = 10404 \end{array}\)

**Verse 2. **Write the following expressions as the square of a sum or a difference:

a.\(4{x^4} + 12{x^2} + 9\)

b.\({x^2}{y^2} – 4xy + 4\)

**Solution guide**

**Question a:**

\(\begin{array}{l} 4{x^4} + 12{x^2} + 9\\ = {(2{x^2})^2} + 2.2{x^2}.3 + {3^2}\\ = {(2{x^2} + 3)^2} \end{array}\)

**Sentence b:**

\(\begin{array}{l} {x^2}{y^2} – 4xy + 4\\ = {(xy)^2} – 2.xy.2 + {2^2}\\ = { (xy – 2)^2} \end{array}\)

**Verse 3.** Collapse the expression:\({(x + y)(x – y)({x^2} + {y^2})}\)

**Solution guide**

\(\begin{array}{*{20}{l}} {(x + y)(x – y)({x^2} + {y^2})}\\ { = \left[ {(x + y)(x – y)} \right]({x^2} + {y^2})}\\ {}\ { = ({x^2} – {y^2})({x^2} + {y^2})}\\ {}\ { = {x^4} – {y^4}} \end{array}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1. **Mental arithmetic:

a. \(999^2\)

b. \(1001^2\)

**Verse 2.** Write the following expressions as the square of a sum or a difference:

a. \(9{x^4} + 12{x^2} + 4\)

b. \({x^2} – 4xy + 4{y^2}\)

**Verse 3. **Collapse the expression: \((2x + y)(2x – y)(4{x^2} + {y^2})\)

### 3.2. Multiple choice exercises

**Question 1: **Maximum value of B=-(2x-3)^{2}+2 is:

A. 1

B. 2

C. 3

D. 4

**Verse 2: **Maximum value of B=(4+x^{2})(4−x^{2}) to be:

A. 12

B. 14

C. 16

D. 18

**Question 3: **Shorten 4x^{2}+2z^{2}−4xz−2z+1 we get the result:

A. \({\left( {2x – z} \right)^2} + {\left( {z – 1} \right)^2}\)

B. \({\left( {2x – z} \right)^2} + {z^2}\)

C. \({\left( {x – 2z} \right)^2} + {\left( {z – 1} \right)^2}\)

D. \({\left( {x – z} \right)^2} + {\left( {z – 1} \right)^2}\)

**Question 4: **Which of the following expressions is positive for all x .?

1. \(x^2+4x+8\)

2.\(x^2+6x+9\)

3.\(x^2-8x+18\)

A. 1

B. 1.2

C. 3

D. 1,2 and 3

**Question 5: **The expression \({x^2} + 4x + 8\) has:

A. GTLN is 8

B. VAT is 4

C. GTLN is 4

D. Net value is 2, GTLN is 8

## 4. Conclusion

Through this lesson, you should know the following:

- Memorize the equality of the square of a sum, the square of a difference, and the difference of two squares.
- Apply the learned equality constants to solve related problems.

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