Question

1. $\frac{50-2w^{2}}{3w^{2}+9w-30} cdot \frac{w^{2}+5w-14}{6w-30}$
2. $\frac{2c^{2}+4c-6}{4c^{2}-7c+3} div \frac{16c^{2}+48c}{16c^{2}-9}$
3. $\frac{16}{3} - \frac{4k+56}{3k+15}$

Explanation:

Problem 5

Step1: Factor all numerators/denominators

Numerator 1: $50-2w^2=-2(w^2-25)=-2(w-5)(w+5)$
Denominator 1: $3w^2+9w-30=3(w^2+3w-10)=3(w+5)(w-2)$
Numerator 2: $w^2+5w-14=(w+7)(w-2)$
Denominator 2: $6w-30=6(w-5)$

Step2: Rewrite product of fractions

$$\frac{-2(w-5)(w+5)}{3(w+5)(w-2)} \cdot \frac{(w+7)(w-2)}{6(w-5)}$$

Step3: Cancel common factors

Cancel $(w-5)$, $(w+5)$, $(w-2)$; simplify constants: $\frac{-2}{3 \cdot 6}=\frac{-2}{18}=\frac{-1}{9}$

Step4: Simplify remaining terms

$$\frac{-1(w+7)}{9}$$

Problem 7

Step1: Factor all numerators/denominators

Numerator 1: $2c^2+4c-6=2(c^2+2c-3)=2(c+3)(c-1)$
Denominator 1: $4c^2-7c+3=(4c-3)(c-1)$
Numerator 2: $16c^2+48c=16c(c+3)$
Denominator 2: $16c^2-9=(4c-3)(4c+3)$

Step2: Rewrite division as multiplication by reciprocal

$$\frac{2(c+3)(c-1)}{(4c-3)(c-1)} \cdot \frac{(4c-3)(4c+3)}{16c(c+3)}$$

Step3: Cancel common factors

Cancel $(c+3)$, $(c-1)$, $(4c-3)$; simplify constants: $\frac{2}{16}=\frac{1}{8}$

Step4: Simplify remaining terms

$$\frac{4c+3}{8c}$$

Problem 9

Step1: Factor denominator and numerator

Denominator 2: $3k+15=3(k+5)$; Numerator 2: $4k+56=4(k+14)$

Step2: Find common denominator $3(k+5)$

$$\frac{16(k+5)}{3(k+5)} - \frac{4(k+14)}{3(k+5)}$$

Step3: Combine fractions

$$\frac{16(k+5)-4(k+14)}{3(k+5)}$$

Step4: Expand and simplify numerator

$16k+80-4k-56=12k+24=12(k+2)$

Step5: Simplify final fraction

$$\frac{12(k+2)}{3(k+5)}=\frac{4(k+2)}{k+5}$$

Answer:

Problem 5: $\frac{-(w+7)}{9}$ or $\frac{-w-7}{9}$
Problem 7: $\frac{4c+3}{8c}$
Problem 9: $\frac{4(k+2)}{k+5}$