Solve the following quadratic equation using the perfect square trinomial pattern: x² + 10x + 25.
To solve the quadratic equation x² + 10x + 25 using the perfect square trinomial pattern, we need to recognize that it can be factored as (x + 5)(x + 5), which represents a perfect square trinomial.
Setting the equation equal to zero, we have (x + 5)(x + 5) = 0.
Using the zero product property, we can conclude that either x + 5 = 0 or x + 5 = 0.
Solving for x in each equation, we find:
x + 5 = 0, so x = -5.
Therefore, the solution to the quadratic equation x² + 10x + 25 = 0 is x = -5.
To solve the quadratic equation x² + 10x + 25 using the perfect square trinomial pattern, follow these steps:
Step 1: Identify the pattern
The pattern for a perfect square trinomial is (a + b)² = a² + 2ab + b².
Step 2: Compare the given equation to the pattern
In our case, the given equation is x² + 10x + 25. We can see that the equation matches the pattern.
Step 3: Apply the pattern
To apply the pattern, we can rewrite our equation as (x + 5)² = x² + 2(5)(x) + 5².
Step 4: Simplify
Using the distributive property in the right side of the equation, we get:
(x + 5)² = x² + 10x + 25.
Step 5: Final step
Now we can see that the given equation is a perfect square trinomial. Therefore, the equation can be factored as (x + 5)².
So, the solution to the quadratic equation x² + 10x + 25 is (x + 5)².
To solve the quadratic equation x² + 10x + 25 using the perfect square trinomial pattern, follow these steps:
Step 1: Identify the pattern
The pattern for a perfect square trinomial is (a + b)² = a² + 2ab + b². In this case, we have x² + 10x + 25, which matches the pattern.
Step 2: Rewrite the equation
Rewrite the original equation using the pattern we identified in step 1. We can rewrite it as (x + 5)² = x² + 2(5)(x) + 5².
Step 3: Simplify the equation
Simplify the equation we obtained in step 2. (x + 5)² = x² + 10x + 25.
Step 4: Equate to zero
Since we want to solve the equation, equate it to zero: (x + 5)² - (x² + 10x + 25) = 0.
Step 5: Expand and simplify
Expand and simplify the equation: x² + 10x + 25 - x² - 10x - 25 = 0.
Step 6: Combine like terms
Combine like terms on both sides of the equation: x² - x² + 10x - 10x + 25 - 25 = 0.
Step 7: Simplify further
Simplify further: 0 = 0.
Step 8: Interpret the result
The equation 0 = 0 is always true, which means that any value of x will satisfy the equation.
Step 9: Final solution
The solution to the quadratic equation x² + 10x + 25 is all real numbers, or x ∈ ℝ.