Solve the following quadratic equation using perfect square trinomial pattern: x^2 - 16x + 64.
A. x = -64
B. x = 64
C. x = -8
D. x = 8
The quadratic equation x^2 - 16x + 64 can be factored using the perfect square trinomial pattern:
(x - 8)^2 = 0
Setting the equation equal to zero:
(x - 8)^2 = 0
Taking the square root of both sides of the equation:
x - 8 = 0
Solving for x:
x = 8
Therefore, the correct answer is D. x = 8.
To solve the quadratic equation x^2 - 16x + 64, we can use the perfect square trinomial pattern.
The perfect square trinomial pattern is given by (a + b)^2 = a^2 + 2ab + b^2, where a is the square root of the first term, and b is half the coefficient of the middle term.
In this case, the first term is x^2, so a = √x^2 = x.
The middle term is -16x, so b = -16/2 = -8.
Now we can rewrite the quadratic equation as (x - 8)^2 = 0.
Setting it equal to zero, we have (x - 8)^2 = 0.
To solve for x, we take the square root of both sides:
√(x - 8)^2 = √0
x - 8 = 0
Adding 8 to both sides, we get:
x = 8
Therefore, the solution to the quadratic equation x^2 - 16x + 64 using the perfect square trinomial pattern is x = 8.
The correct answer is D. x = 8.
To solve the quadratic equation x^2 - 16x + 64 using the perfect square trinomial pattern, we need to check if the equation can be factored as a perfect square.
The perfect square trinomial pattern is in the form (x - a)^2 = x^2 - 2ax + a^2.
Comparing the given quadratic equation x^2 - 16x + 64 with the perfect square trinomial pattern, we observe that the square root of the constant term (64) is 8, and the coefficient of the middle term (-16) is double the product of the square root of the constant term and the coefficient of x, which is -16 = -2 * (8 * 1).
Hence, we can write the given quadratic equation as (x - 8)^2 = 0.
Now, setting the expression (x - 8)^2 to zero, we can solve for x:
(x - 8)^2 = 0
Taking the square root of both sides:
x - 8 = 0
Adding 8 to both sides:
x = 8
Therefore, the solution to the quadratic equation x^2 - 16x + 64 is x = 8.
So, the correct answer is option D: x = 8.