Solve the following quadratic equation using the perfect square trinomial pattern: 1/4 x^2 + x + 4
To solve the quadratic equation 1/4x^2 + x + 4 = 0 using the perfect square trinomial pattern, we need to rewrite the equation in the form (ax + b)^2 = 0.
First, we can factor out 1/4 from the quadratic equation:
1/4x^2 + x + 4 = (1/4)(x^2 + 4x + 16)
Next, we want to find the perfect square trinomial that is equivalent to x^2 + 4x + 16:
The middle coefficient, 4, is twice the product of the square root of the first and last terms, which are x and 16, respectively. Therefore, 4 is equal to 2 * √(x * 16) = 2 * 4 * √x = 8√x.
Using this information, we can rewrite the perfect square trinomial as:
x^2 + 4x + 16 = (x + 2√x)^2
Now, we can go back to the factored equation and substitute (x + 2√x)^2 for x^2 + 4x + 16:
(1/4)(x + 2√x)^2 = 0
Finally, we can solve for x by setting the factored equation equal to zero:
(x + 2√x)^2 = 0
Taking the square root of both sides:
x + 2√x = 0
Subtracting 2√x from both sides:
x = -2√x
To solve this equation, we can square both sides:
x^2 = 4x
Rearranging and subtracting 4x from both sides:
x^2 - 4x = 0
Factoring:
x(x - 4) = 0
Setting each factor equal to zero:
x = 0 or x - 4 = 0
Solving for x, we get two solutions:
x = 0, or x = 4
Therefore, the quadratic equation 1/4x^2 + x + 4 = 0 has two solutions: x = 0 and x = 4.