(x%5E2-4x%2B4)%3D0.jpeg "(x^2+6x+9)(x^2-4x+4)=0")
Solving the Equation: (x^2 + 6x + 9)(x^2 - 4x + 4) = 0
This equation presents a unique opportunity to showcase the power of factoring and the zero product property. Let's break down the solution step by step:
Recognizing the Patterns
Notice that both expressions within the parentheses are perfect square trinomials.
- x^2 + 6x + 9: This factors into (x + 3)^2
- x^2 - 4x + 4: This factors into (x - 2)^2
Applying the Zero Product Property
Now our equation becomes:
(x + 3)^2 (x - 2)^2 = 0
The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Therefore, we can set each factor equal to zero and solve:
- (x + 3)^2 = 0
- Taking the square root of both sides, we get x + 3 = 0
- Solving for x, we get x = -3
- (x - 2)^2 = 0
- Taking the square root of both sides, we get x - 2 = 0
- Solving for x, we get x = 2
The Solutions
We have found two solutions to the equation:
- x = -3
- x = 2
Both solutions are double roots because they appear twice in the factored form of the equation. This means each solution has a multiplicity of 2.
Understanding Multiplicity
The multiplicity of a root tells us how many times a particular root appears in the factored form of a polynomial. In this case, the double roots indicate that the graph of the polynomial touches the x-axis at both x = -3 and x = 2, but does not cross it.
By recognizing the patterns in the equation and applying the zero product property, we have successfully solved the equation and gained insights into the behavior of the polynomial's graph.