Find the factors of the following numbers:
1. 100
2. 30
3. 12
4. 24
5. 15
Solve the following quadratic equations by factoring.
1. [tex]$-3 m^2+27 m=0$[/tex]
2. [tex]$x^2-2 x=15$[/tex]
Answer (1)
Factors: 100: 1, 2, 4, 5, 10, 20, 25, 50, 100; 30: 1, 2, 3, 5, 6, 10, 15, 30; 12: 1, 2, 3, 4, 6, 12; 24: 1, 2, 3, 4, 6, 8, 12, 24; 15: 1, 3, 5, 15. Solutions: m = 0 , 9 ; x = 5 , − 3
Explanation
Problem Analysis We are asked to find the factors of the numbers 100, 30, 12, 24, and 15, and to solve two quadratic equations by factoring.
Finding Factors The factors of a number are the integers that divide evenly into that number. We can find these by checking which numbers from 1 to the number itself divide evenly.
Factors of 100 The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
Factors of 30 The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
Factors of 12 The factors of 12 are 1, 2, 3, 4, 6, and 12.
Factors of 24 The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Factors of 15 The factors of 15 are 1, 3, 5, and 15.
Solving Quadratic Equations Now, let's solve the quadratic equations by factoring.
Solving -3m^2 + 27m = 0
− 3 m 2 + 27 m = 0 . We can factor out a − 3 m from both terms: − 3 m ( m − 9 ) = 0 Setting each factor to zero gives us: − 3 m = 0 ⇒ m = 0 m − 9 = 0 ⇒ m = 9 So the solutions are m = 0 and m = 9 .
Solving x^2 - 2x = 15
x 2 − 2 x = 15 . First, we rewrite the equation in standard form: x 2 − 2 x − 15 = 0 Now we look for two numbers that multiply to -15 and add to -2. These numbers are -5 and 3. So we can factor the quadratic as: ( x − 5 ) ( x + 3 ) = 0 Setting each factor to zero gives us: x − 5 = 0 ⇒ x = 5 x + 3 = 0 ⇒ x = − 3 So the solutions are x = 5 and x = − 3 .
Final Answer Therefore, the factors of the given numbers are:
100: 1, 2, 4, 5, 10, 20, 25, 50, 100
30: 1, 2, 3, 5, 6, 10, 15, 30
12: 1, 2, 3, 4, 6, 12
24: 1, 2, 3, 4, 6, 8, 12, 24
15: 1, 3, 5, 15
The solutions to the quadratic equations are:
− 3 m 2 + 27 m = 0 : m = 0 , 9
x 2 − 2 x = 15 : x = 5 , − 3
Examples
Factoring numbers is a fundamental concept in mathematics with various real-life applications. For instance, when dividing a group of people into teams, you need to find the factors of the total number of people to determine the possible team sizes. Similarly, factoring quadratic equations is crucial in physics to solve problems related to projectile motion or in engineering to design structures. Understanding these concepts helps in optimizing resources and making informed decisions in various fields.
