3x^4 + 5x^2 - 2 = 0
give imaginary and real roots
rational roots theorem factors of (+-)p/q are possible rational zeros of function f where the coefficients of f are integers.
how do you go about solving this?
let y = x2, so
so 3x^4 + 5x^2 - 2 = 0 ---> 3y^2 + 5y - 2 = 0
(3y - 1)(y + 2) = 0
y = 1/3 or y = -2
so
x^2 = 1/2 OR x^2 = -2
x = ±1/√2 or x = ± i√2
can be shortened if you can see that
3x^4 + 5x^2 - 2 = 0 ---> (3x^2 - 1)(x^2 + 2) = 0
etc
To solve the given equation, 3x^4 + 5x^2 - 2 = 0, you can use factoring, the rational roots theorem, and the quadratic formula.
Step 1: Look for any common factors to factor out. In this case, there are no common factors among all three terms.
Step 2: Apply the rational roots theorem to find the possible rational roots. The rational roots theorem states that any rational roots of a polynomial equation are of the form p/q, where p is a factor of the constant term (-2 in this case) and q is a factor of the leading coefficient (3 in this case). The factors of -2 are {-2, -1, 1, 2}, and the factors of 3 are {-3, -1, 1, 3}. Therefore, the possible rational roots are:
±1/3, ±2/3, ±1/1, ±2/1
Step 3: Use the synthetic division or long division method to test each of the possible rational roots obtained in Step 2. By trying out each potential root, we can find the actual roots of the equation.
However, in this case, using the rational roots theorem and testing the possible rational roots of the equation will not give you any real or rational roots. So, we need to proceed using the quadratic formula to find the roots.
Step 4: Use the quadratic formula, which states that the solutions of a quadratic equation of the form ax^2 + bx + c = 0, can be found using the formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In our case, a = 3, b = 5, and c = -2. Plugging these values into the quadratic formula:
x = (-5 ± √(5^2 - 4 * 3 * (-2))) / (2 * 3)
x = (-5 ± √(25 + 24)) / 6
x = (-5 ± √49) / 6
Step 5: Simplify the equation further:
x = (-5 ± 7) / 6
Now, there are two possible solutions:
x = (-5 + 7) / 6 = 2 / 6 = 1/3 (real root)
x = (-5 - 7) / 6 = -12 / 6 = -2 (real root)
Therefore, the equation 3x^4 + 5x^2 - 2 = 0 has two real roots: x = 1/3 and x = -2.