16x^4 + 65x^2 + 4 = 0
(16x^2+1)(x^2+4) = 0
No real roots, but easy-to-find complex roots
To find the solutions of the equation 16x^4 + 65x^2 + 4 = 0, we can use a method called factoring or the quadratic formula. In this case, we have a quadratic equation in terms of x^2, so we can solve it by treating x^2 as a variable.
Let's start by factoring the equation. Factoring is the process of expressing a polynomial as the product of its factors. In this case, our equation has the form Ax^2 + Bx + C = 0, where A, B, and C are constants.
Factorizing the equation:
16x^4 + 65x^2 + 4 = 0
We can rewrite the equation as:
(4x^2 + 1)^2 + 64x^2 = 0
Let's define a new variable:
y = 4x^2 + 1
Now, substituting y back into the equation, we get:
y^2 + 64x^2 = 0
This equation can be further factorized:
(y + 8x)(y - 8x) = 0
Now, let's substitute the value of y back into the equation to solve for x:
(4x^2 + 1 + 8x)(4x^2 + 1 - 8x) = 0
Now, solving each factor:
4x^2 + 1 + 8x = 0
and
4x^2 + 1 - 8x = 0
Solving the first equation:
4x^2 + 8x + 1 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, the quadratic equation is in the form ax^2 + bx + c = 0, where:
a = 4, b = 8, c = 1
Calculating the discriminant (b^2 - 4ac):
D = (8^2) - (4 * 4 * 1) = 64 - 16 = 48
Since the discriminant is positive, we have two real solutions for x:
x = (-8 ± sqrt(48)) / (2 * 4)
Simplifying:
x = (-8 ± sqrt(48)) / 8
Further simplifying:
x = (-8 ± 4 * sqrt(3)) / 8
Reducing the fraction:
x = -1 ± (1/2) * sqrt(3)
So, the solutions of the equation 16x^4 + 65x^2 + 4 = 0 are:
x = -1 + (1/2) * sqrt(3)
x = -1 - (1/2) * sqrt(3)