If the product of two or more factors is zero then at least one of the factors should be zero. This property is known as Zero-Product Property.
That is, if XY = 0, then X = 0 or Y = 0 or both X and Y are equal to 0.
Similarly, if XYZ = 0, then X = 0 or Y = 0 or Z = 0 or all three are 0.
This thought can be extended to any number of factors.
The x-coordinate of the point where a line intersects the y-axis is 0. So, the y-intercept of a line can also be found by substituting x = 0 in the equation of the line.
Let's solve this equation through factoring, followed by the application of the Zero-Product Property
On factoring,
x2-4x = 0 ⇒ x (4 - x) = 0 [Taking out the common variable x]
⇒ x = 0 or (4 - x) = 0 [Recall the definition of Zero-product Property.]
⇒ x = 0 or x = 4
So, the solutions are x = 0 or x = 4
Both these values satisfy the original equation x2 - 4x = 0
x2 + 4x - 5 = 0
⇒ x2 + 5x - x - 5 = 0 [Splitting the middle term]
⇒ x(x + 5) - 1(x+5) = 0 [Taking out the common factors]
(x + 5)(x - 1) = 0 [Factor.]
⇒ x + 5 = 0 or (x - 1) = 0 [Recall the definition of Zero-product property.]
⇒ x = -5 or x = 1
So, the solutions are x = -5 or x = 1
Both these values satisfy the original equation x2 + 4x - 5 = 0.
Q1: Solve for x: (x-2)(x+3) = 0
Q2: Solve for x: x(x-5) = 0
Q: When can I use the Zero-Product Property?
A: You can use the Zero-Product Property when you have an equation where a product of factors equals zero.
Q: What if none of the factors equal zero?
A: If none of the factors equal zero, then the product cannot equal zero. Thus, there are no solutions.