Product equation is an expression of the form: a * b = 0, where The and B they are algebraic terms. The resolution must be based on the following property of real numbers:
If a = 0 or b = 0, we have to a * b = 0.
if a*b, then a = 0 and b = 0
We will, through practical examples, demonstrate the ways to solve a product equation, based on the property presented above.
the equation (x + 2) * (2x + 6) = 0 can be considered a product equation because:
(x + 2) = 0 → x + 2 = 0 → x = –2
(2x + 6) = 0 → 2x + 6 = 0 → 2x = –6 → x = –3
For x + 2 = 0, we have x = –2 and for 2x + 6 = 0, we have x = –3.
Take another example:
(4x - 5) * (6x - 2) = 0
4x – 5 = 0 → 4x = 5 → x = 5/4
6x – 2 = 0 → 6x = 2 → x = 2/6 → x = 1/3
For 4x – 5 = 0, we have x = 5/4 and for 6x – 2 = 0, we have x = 1/3
The product equations can be solved in other ways, it will depend on how they are presented. In many cases, resolution is only possible using factorization.
Example 1
4x² - 100 = 0
The equation presented is called the difference between two squares and can be written as a product of the sum and the difference: (2x – 10) * (2x + 10) = 0. Track the resolution after factoring:
(2x – 10) * (2x + 10) = 0
2x – 10 = 10 → 2x = 10 → x = 10/2 → x’ = 5
2x + 10 = 0 → 2x = –10 → x = –10/2 → x’’ = – 5
Another form of resolution would be:
4x² - 100 = 0
4x² = 100
x² = 100/4
x² = 25
√x² = √25
x’ = 5
x’’ = – 5
Example 2
x² + 6x + 9 = 0
By factoring the 1st member of the equation, we have (x + 3)². Then:
(x + 3)² = 0
x + 3 = 0
x = – 3
Example 3
18x² + 12x = 0
Let's use common factor factoring in evidence.
6x * (3x + 2) = 0
6x = 0
x = 0/6
x’ = 0
3x + 2 = 0
3x = –2
x’’ = –2/3
Do not stop now... There's more after the advertising ;)
by Mark Noah
Graduated in Mathematics
Brazil School Team
Equation - Math - Brazil School
Would you like to reference this text in a school or academic work? Look:
SILVA, Marcos Noé Pedro da. "Product Equation Solving"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/resolucao-equacao-produto.htm. Accessed on June 29, 2021.
