Question

topic: exploring the density of the number line. find three numbers that are between the two given numbers. 1. $5\frac{3}{4}$ and $6\frac{1}{3}$ 2. $-2\frac{1}{4}$ and $-1\frac{1}{2}$ 3. $\frac{1}{4}$ and $\frac{5}{8}$ 4. $\sqrt{3}$ and $\sqrt{5}$ 5. 4 and $\sqrt{23}$ 6. $-9\frac{3}{4}$ and - 8.5 7. $\sqrt{\frac{1}{4}}$ and $\sqrt{\frac{4}{9}}$ 8. $\sqrt{13}$ and $\sqrt{14}$ set topic: factoring quadratics the area of a rectangle is given in the form of a trinomial expression. find the equivalent expression that shows the lengths of the two sides of the rectangle. 9. $x^{2}+9x + 8$ 10. $x^{2}-6x + 8$ 11. $x^{2}-2x - 8$ 12. $x^{2}+7x - 8$ 13. $x^{2}-11x + 24$ 14. $x^{2}-14x + 24$ 15. $x^{2}-25x + 24$ 16. $x^{2}-10x + 24$ 17. $x^{2}-2x - 24$ 18. $x^{2}-5x - 24$ 19. $x^{2}+5x - 24$ 20. $x^{2}-10x + 25$ 21. $x^{2}-25$ 22. $x^{2}-2x - 15$ 23. $x^{2}+10x - 75$ 24. $x^{2}-20x + 51$ 25. $x^{2}+14x - 32$ 26. $x^{2}-1$ 27. $x^{2}-2x + 1$ 28. $x^{2}+12x - 45$

Explanation:

1. For finding numbers between $5\frac{3}{4}$ and $6\frac{1}{3}$:

First, convert to improper - fractions. $5\frac{3}{4}=\frac{23}{4}$ and $6\frac{1}{3}=\frac{19}{3}$. Find a common denominator, which is 12. $\frac{23}{4}=\frac{69}{12}$ and $\frac{19}{3}=\frac{76}{12}$. Three numbers between them could be $\frac{70}{12}=\frac{35}{6}$, $\frac{72}{12} = 6$, $\frac{74}{12}=\frac{37}{6}$.

2. For finding numbers between $-2\frac{1}{4}$ and $-1\frac{1}{2}$:

Convert to improper - fractions. $-2\frac{1}{4}=-\frac{9}{4}$ and $-1\frac{1}{2}=-\frac{3}{2}$. The common denominator is 4, and $-\frac{3}{2}=-\frac{6}{4}$. Three numbers between them could be $-\frac{7}{4}$, $-\frac{3}{2}=-\frac{6}{4}$, $-\frac{5}{4}$.

3. For finding numbers between $\frac{1}{4}$ and $\frac{5}{8}$:

The common denominator is 8. $\frac{1}{4}=\frac{2}{8}$. Three numbers between them could be $\frac{3}{8}$, $\frac{4}{8}=\frac{1}{2}$, $\frac{5}{8}$.

4. For finding numbers between $\sqrt{3}\approx1.732$ and $\sqrt{5}\approx2.236$:

Three numbers between them could be $1.8$, $2$, $2.2$.

5. For finding numbers between 4 and $\sqrt{23}\approx4.796$:

Three numbers between them could be $4.1$, $4.5$, $4.7$.

6. For finding numbers between $-9\frac{3}{4}=-\frac{39}{4}=-9.75$ and $-8.5$:

Three numbers between them could be $-9.5$, $-9$, $-8.75$.

7. For finding numbers between $\sqrt{\frac{1}{4}}=\frac{1}{2}$ and $\sqrt{\frac{4}{9}}=\frac{2}{3}$:

The common denominator is 6. $\frac{1}{2}=\frac{3}{6}$ and $\frac{2}{3}=\frac{4}{6}$. Three numbers between them could be $\frac{7}{12}$, $\frac{2}{3}=\frac{8}{12}$, $\frac{3}{4}=\frac{9}{12}$.

8. For finding numbers between $\sqrt{13}\approx3.606$ and $\sqrt{14}\approx3.742$:

Three numbers between them could be $3.62$, $3.65$, $3.7$.

9. For factoring $x^{2}+9x + 8$:

We need to find two numbers that multiply to 8 and add up to 9. The numbers are 1 and 8. So $x^{2}+9x + 8=(x + 1)(x + 8)$.

10. For factoring $x^{2}-6x + 8$:

We find two numbers that multiply to 8 and add up to - 6. The numbers are - 2 and - 4. So $x^{2}-6x + 8=(x - 2)(x - 4)$.

11. For factoring $x^{2}-2x - 8$:

We find two numbers that multiply to - 8 and add up to - 2. The numbers are - 4 and 2. So $x^{2}-2x - 8=(x - 4)(x+2)$.

12. For factoring $x^{2}+7x - 8$:

We find two numbers that multiply to - 8 and add up to 7. The numbers are 8 and - 1. So $x^{2}+7x - 8=(x + 8)(x - 1)$.

13. For factoring $x^{2}-11x + 24$:

We find two numbers that multiply to 24 and add up to - 11. The numbers are - 3 and - 8. So $x^{2}-11x + 24=(x - 3)(x - 8)$.

14. For factoring $x^{2}-14x + 24$:

We find two numbers that multiply to 24 and add up to - 14. The numbers are - 2 and - 12. So $x^{2}-14x + 24=(x - 2)(x - 12)$.

15. For factoring $x^{2}-25x + 24$:

We find two numbers that multiply to 24 and add up to - 25. The numbers are - 1 and - 24. So $x^{2}-25x + 24=(x - 1)(x - 24)$.

16. For factoring $x^{2}-10x + 24$:

We find two numbers that multiply to 24 and add up to - 10. The numbers are - 4 and - 6. So $x^{2}-10x + 24=(x - 4)(x - 6)$.

17. For factoring $x^{2}-2x - 24$:

We find two numbers that multiply to - 24 and add up to - 2. The numbers are - 6 and 4. So $x^{2}-2x - 24=(x - 6)(x + 4)$.

18. For factoring $x^{2}-5x - 24$:

We find two numbers that multiply to - 24 and add up to - 5. The numbers are - 8 and 3. So $x^{2}-5x - 24=(x - 8)(x+3)$.

19. For factoring $x^{2}+5x - 24$:

We find two numbers that multiply to - 24 and add up to 5. The numbers are 8 and - 3. So $x^{2}+5x - 24=(x + 8)(x - 3)$.

20. For factoring $x^{2}-10x + 25$:

This is a perfect - square trinomial. $x^{2}-10x + 25=(x - 5)^{2}$.

21. For factoring $x^{2}-25$:

Using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$, where $a = x$ and $b = 5$. So $x^{2}-25=(x + 5)(x - 5)$.

22. For factoring $x^{2}-2x - 15$:

We find two numbers that multiply to - 15 and add up to - 2. The numbers are - 5 and 3. So $x^{2}-2x - 15=(x - 5)(x+3)$.

23. For factoring $x^{2}+10x - 75$:

We find two numbers that multiply to - 75 and add up to 10. The numbers are 15 and - 5. So $x^{2}+10x - 75=(x + 15)(x - 5)$.

24. For factoring $x^{2}-20x + 51$:

We find two numbers that multiply to 51 and add up to - 20. The numbers are - 17 and - 3. So $x^{2}-20x + 51=(x - 17)(x - 3)$.

25. For factoring $x^{2}+14x - 32$:

We find two numbers that multiply to - 32 and add up to 14. The numbers are 16 and - 2. So $x^{2}+14x - 32=(x + 16)(x - 2)$.

26. For factoring $x^{2}-1$:

Using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$, where $a = x$ and $b = 1$. So $x^{2}-1=(x + 1)(x - 1)$.

27. For factoring $x^{2}-2x + 1$:

This is a perfect - square trinomial. $x^{2}-2x + 1=(x - 1)^{2}$.

28. For factoring $x^{2}+12x - 45$:

We find two numbers that multiply to - 45 and add up to 12. The numbers are 15 and - 3. So $x^{2}+12x - 45=(x + 15)(x - 3)$.

Answer:

1. $\frac{35}{6}$, $6$, $\frac{37}{6}$
2. $-\frac{7}{4}$, $-\frac{3}{2}$, $-\frac{5}{4}$
3. $\frac{3}{8}$, $\frac{1}{2}$, $\frac{5}{8}$
4. $1.8$, $2$, $2.2$
5. $4.1$, $4.5$, $4.7$
6. $-9.5$, $-9$, $-8.75$
7. $\frac{7}{12}$, $\frac{2}{3}$, $\frac{3}{4}$
8. $3.62$, $3.65$, $3.7$
9. $(x + 1)(x + 8)$
10. $(x - 2)(x - 4)$
11. $(x - 4)(x + 2)$
12. $(x + 8)(x - 1)$
13. $(x - 3)(x - 8)$
14. $(x - 2)(x - 12)$
15. $(x - 1)(x - 24)$
16. $(x - 4)(x - 6)$
17. $(x - 6)(x + 4)$
18. $(x - 8)(x + 3)$
19. $(x + 8)(x - 3)$
20. $(x - 5)^{2}$
21. $(x + 5)(x - 5)$
22. $(x - 5)(x + 3)$
23. $(x + 15)(x - 5)$
24. $(x - 17)(x - 3)$
25. $(x + 16)(x - 2)$
26. $(x + 1)(x - 1)$
27. $(x - 1)^{2}$
28. $(x + 15)(x - 3)$

Answer:

1. $\frac{35}{6}$, $6$, $\frac{37}{6}$
2. $-\frac{7}{4}$, $-\frac{3}{2}$, $-\frac{5}{4}$
3. $\frac{3}{8}$, $\frac{1}{2}$, $\frac{5}{8}$
4. $1.8$, $2$, $2.2$
5. $4.1$, $4.5$, $4.7$
6. $-9.5$, $-9$, $-8.75$
7. $\frac{7}{12}$, $\frac{2}{3}$, $\frac{3}{4}$
8. $3.62$, $3.65$, $3.7$
9. $(x + 1)(x + 8)$
10. $(x - 2)(x - 4)$
11. $(x - 4)(x + 2)$
12. $(x + 8)(x - 1)$
13. $(x - 3)(x - 8)$
14. $(x - 2)(x - 12)$
15. $(x - 1)(x - 24)$
16. $(x - 4)(x - 6)$
17. $(x - 6)(x + 4)$
18. $(x - 8)(x + 3)$
19. $(x + 8)(x - 3)$
20. $(x - 5)^{2}$
21. $(x + 5)(x - 5)$
22. $(x - 5)(x + 3)$
23. $(x + 15)(x - 5)$
24. $(x - 17)(x - 3)$
25. $(x + 16)(x - 2)$
26. $(x + 1)(x - 1)$
27. $(x - 1)^{2}$
28. $(x + 15)(x - 3)$