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]

The factors for the provided numbers are identified, while the quadratic equations are solved by factoring. For the first quadratic, the solutions are m = 0 and m = 9, and for the second, the solutions are x = 5 and x = -3.
;

Answered by Anonymous | 2025-07-08

Find the factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.
Find the factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
Find the factors of 12: 1, 2, 3, 4, 6, 12.
Find the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Find the factors of 15: 1, 3, 5, 15.
Solve − 3 m 2 + 27 m = 0 by factoring: m = 0 , 9 .
Solve x 2 − 2 x = 15 by factoring: x = 5 , − 3 .

Explanation

Finding Factors We are asked to find the factors of the numbers 100, 30, 12, 24, and 15, and to solve two quadratic equations by factoring. Let's start by listing the factors of each number.

Factors of 100 The factors of 100 are the numbers that divide 100 evenly. These are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

Factors of 30 The factors of 30 are the numbers that divide 30 evenly. These are 1, 2, 3, 5, 6, 10, 15, and 30.

Factors of 12 The factors of 12 are the numbers that divide 12 evenly. These are 1, 2, 3, 4, 6, and 12.

Factors of 24 The factors of 24 are the numbers that divide 24 evenly. These are 1, 2, 3, 4, 6, 8, 12, and 24.

Factors of 15 The factors of 15 are the numbers that divide 15 evenly. These are 1, 3, 5, and 15.

Solving the First Quadratic Equation Now, let's solve the first quadratic equation: − 3 m 2 + 27 m = 0 . We can factor out a common factor of − 3 m from both terms: − 3 m ( m − 9 ) = 0 .

Solutions for m Setting each factor to zero, we have − 3 m = 0 or m − 9 = 0 . Solving for m , we get m = 0 or m = 9 . Thus, the solutions are m = 0 and m = 9 .

Solving the Second Quadratic Equation Next, let's solve the second quadratic equation: x 2 − 2 x = 15 . First, we rewrite the equation in the standard form a x 2 + b x + c = 0 : x 2 − 2 x − 15 = 0 .

Factoring the Quadratic Now, we factor the quadratic expression: ( x − 5 ) ( x + 3 ) = 0 .

Solutions for x Setting each factor to zero, we have x − 5 = 0 or x + 3 = 0 . Solving for x , we get x = 5 or x = − 3 . Thus, the solutions are x = 5 and x = − 3 .

Final Answer In summary: Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 15: 1, 3, 5, 15 Solutions to − 3 m 2 + 27 m = 0 : m = 0 , 9 Solutions to x 2 − 2 x = 15 : x = 5 , − 3


Examples
Understanding factors and solving quadratic equations are fundamental concepts in algebra. For instance, imagine you are designing a rectangular garden with an area of 100 square feet. The factors of 100 (1, 2, 4, 5, 10, 20, 25, 50, 100) represent the possible dimensions (length and width) of the garden. Similarly, solving quadratic equations can help determine the dimensions needed to enclose a specific area with a given amount of fencing, optimizing space and resources in real-world applications.

Answered by GinnyAnswer | 2025-07-08