Solve the following quadratic equation using the perfect square trinomial pattern: x² - 16x +64.
The perfect square trinomial pattern is (a - b)² = a² - 2ab + b².
In this case, a = x and b = 8.
We can rewrite the equation as:
x² - 16x + 64 = (x - 8)²
Therefore, the solution to the quadratic equation x² - 16x + 64 = 0 is x = 8.
To solve the quadratic equation x² - 16x + 64 using the perfect square trinomial pattern, follow these steps:
Step 1: Determine if the equation fits the perfect square trinomial pattern.
The perfect square trinomial pattern can be written as (a - b)² = a² - 2ab + b².
In this case, the quadratic equation x² - 16x + 64 has the form a² - 2ab + b², so it fits the perfect square trinomial pattern.
Step 2: Identify the values of the variables.
In the equation x² - 16x + 64, we have:
a = x
b = 8
Step 3: Write the perfect square trinomial.
Based on the values of a and b, we can write the perfect square trinomial as (x - 8)².
Step 4: Solve for x.
To solve for x, take the square root of both sides of the equation, remembering the ± sign:
√[(x - 8)²] = ±√64
Simplifying,
x - 8 = ±8
Now, solve for x in both cases:
Case 1: x - 8 = 8
x = 8 + 8
x = 16
Case 2: x - 8 = -8
x = -8 + 8
x = 0
Therefore, the solutions to the quadratic equation x² - 16x + 64 are x = 16 and x = 0.
To solve the quadratic equation x² - 16x + 64 using the perfect square trinomial pattern, follow these steps:
Step 1: Identify the quadratic equation in the form ax² + bx + c.
In this case, a = 1, b = -16, and c = 64.
Step 2: Check if the quadratic equation can be factored using the perfect square trinomial pattern. For this pattern to apply, the equation should have the form (x - a)² = 0.
Step 3: Rewrite the equation in the form (x - a)² = 0.
To rewrite the equation x² - 16x + 64, find the value of "a" that satisfies the equation (x - a)² = x² - 16x + 64.
In this case, "a" would be half of the coefficient of the linear term, which is (-16/2 = -8). So, we have:
(x - (-8))² = 0
(x + 8)² = 0
Step 4: Solve for x by taking the square root of both sides of the equation.
Taking the square root of (x + 8)² = 0, we have:
√((x + 8)²) = ±√0
(x + 8) = ±0
Simplifying further, we have:
x + 8 = 0 (if using the positive square root)
x + 8 = 0 (if using the negative square root)
Step 5: Solve for x.
For x + 8 = 0, subtracting 8 from both sides yields:
x = -8
Therefore, the solution to the quadratic equation x² - 16x + 64 = 0 is x = -8.