# Uses of Factorization - Examples, Exercises and Solutions

Factorization is the main key to solving more complex exercises than those you have studied up to today.
Factorization helps to solve different exercises, among them, those that have algebraic fractions.
In exercises where the sum or difference of their terms equals zero, factorization allows us to see them as a multiplication of $0$ and thus discover the terms that lead them to this result.

For exercises composed of fractions with expressions that may seem complicated, we can break them down into factors, reduce them, and thus end up with much simpler fractions.

## Practice Uses of Factorization

### Exercise #1

Find the value of the parameter x.

$-x^2-7x-12=0$

### Video Solution

$x=-3,x=-4$

### Exercise #2

Find the value of the parameter x.

$2x^2-7x+5=0$

### Video Solution

$x=1,x=2.5$

### Exercise #3

Find the value of the parameter x.

$x^2-25=0$

### Video Solution

$x=5,x=-5$

### Exercise #4

Find the value of the parameter x.

$(x-5)^2=0$

### Video Solution

$x=5$

### Exercise #5

Find the value of the parameter x.

$(x+5)^2=0$

### Video Solution

$x=-5$

### Exercise #1

Find the value of the parameter x.

$12x^3-9x^2-3x=0$

### Video Solution

$x=0,x=1,x=-\frac{1}{4}$

### Exercise #2

Find the value of the parameter x.

$(x-4)^2+x(x-12)=16$

### Video Solution

$x=0,x=10$

### Exercise #3

Find the value of the parameter x.

$-2x(3-x)+(x-3)^2=9$

### Video Solution

$x=0,x=4$

### Exercise #4

A right triangle is shown below.

x>1

Find the lengths of the sides of the triangle.

### Video Solution

$6,8,10$

### Exercise #5

A right triangle is shown below.

x>1

Calculate the lengths of the sides of the triangle.

### Video Solution

$5,12,13$

### Exercise #1

In front of you is a square.

The expressions listed next to the sides describe their length.

( x>-2 length measurements in cm).

Since the area of the square is 16.

Find the lengths of the sides of the square.

4

### Exercise #2

In front of you is a square.

The expressions listed next to the sides describe their length.

( x>-4 length measurements in cm).

Since the area of the square is 36.

Find the lengths of the sides of the square.

6

### Exercise #3

In front of you is an isosceles right triangle.

The expressions listed next to the sides describe their length.

( x>-5 length measurements in cm).

Since the area of the triangle is 12.5.

Find the lengths of the sides of the triangle.

### Video Solution

$5,5,5\sqrt{2}$

### Exercise #4

In front of you is an isosceles right triangle.

The expressions listed next to the sides describe their length.

( x>-8 length measurements in cm).

Since the area of the triangle is 32.

Find the lengths of the sides of the triangle.

### Video Solution

$8,8,8\sqrt{2}$

### Topics learned in later sections

1. Solving Equations by Factoring