Solve the equation x^3 - 2x + 4 = 0.
==> I know the answer is -2, but I can't seem to remember how to solve these by hand without using a graphing calculator. Any help is greatly appreciated!! :)
Unless they factor, cubic equations cannot be easily solved "by hand"
In this case, try subbing ±1 and ±2 into the equation to get a zero.
You were right to find that x = -2 works
so (x+2) must be a factor.
Use long algebraic division or synthetic division to find the other factor to be (x^2 - 2x + 2)
using the quadratic formula on
x^2 - 2x + 2 = 0 gives me two complex roots, namely (1 ± i)/2
the two complex roots should have been just
1 ± i
Ohhh I see. Thanks Reiny this helped a lot!! :)
To solve the equation x^3 - 2x + 4 = 0, we can use the rational root theorem combined with synthetic division or long division.
Step 1: Use the rational root theorem to determine possible rational roots. The rational root theorem states that if the equation has a rational root p/q, where p is a factor of the constant term (4 in this case) and q is a factor of the leading coefficient (1 in this case), then p/q will be a root of the equation. In this case, the factors of 4 are ±1, ±2, ±4 and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±2, and ±4.
Step 2: Begin testing the possible rational roots by substituting them one by one into the original equation. We'll first check x = -1:
(-1)^3 - 2(-1) + 4 = -1 + 2 + 4 = 5
Since this does not satisfy the equation, -1 is not a root.
Step 3: Continue testing the remaining possible rational roots. We'll now check x = -2:
(-2)^3 - 2(-2) + 4 = -8 + 4 + 4 = 0
Since this satisfies the equation, -2 is a root.
Step 4: Divide the equation by (x - (-2)), or (x + 2), using either synthetic division or long division. Let's use synthetic division:
-2 | 1 0 -2 4
-----------------
-2 4 -4
-----------
1 -2 0
The result of the division is the quotient 1x^2 - 2x + 0. Notice that the last term, -4, is now eliminated.
Step 5: Now, we have a quadratic equation, 1x^2 - 2x = 0, which can be factored as x(x - 2) = 0. From this equation, we can see that x = 0 or x = 2.
Therefore, the solutions to the equation x^3 - 2x + 4 = 0 are x = -2, x = 0, and x = 2.