-x³ + 4x = mx Rearranged as: x³ - 4x + mx = 0 Factored: x(x² - 4 + m) = 0 Using the quadratic formula for x² - 4 + m = 0: x = \frac{0 \pm \sqrt{0 - 4(1)(m - 4)}}{2(1)} = \frac{\pm \sqrt{-4m + 16}}{2} = \pm \frac{\sqrt{4 - m}}{1} = \pm \sqrt{4 - m} Therefore, the solutions are: x = 0, \pm \sqrt{4 - m}
Answer (2)
The solutions to the equation − x 3 + 4 x = m x are found by rearranging and factoring to derive x = 0 and x = ± 4 − m . The quadratic formula is used for finding the additional solutions based on the value of m . Thus, the complete solutions are x = 0 , ± 4 − m .
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To solve the equation − x 3 + 4 x = m x , we first rearrange it to the form x 3 − 4 x + m x = 0 .
This allows us to rewrite the equation as:
x ( x 2 − 4 + m ) = 0
From this factored form, we can see that one solution is clearly x = 0 .
Next, we need to solve for x in the quadratic equation x 2 − 4 + m = 0 .
Using the quadratic formula, where a = 1 , b = 0 , and c = − 4 + m , we have:
x = 2 a − b ± b 2 − 4 a c
Substituting in the values gives:
x = 2 ( 1 ) 0 ± 0 − 4 ( 1 ) ( − 4 + m ) = 2 ± 16 − 4 m
Simplifying this expression results in:
x = ± 4 − m
Thus, the solutions to the equation − x 3 + 4 x = m x are:
x = 0
x = 4 − m
x = − 4 − m
These solutions are valid as long as the expression 4 − m yields a non-negative number, ensuring real solutions for the square root. If 4"> m > 4 , the solutions involving the square root become complex, and only x = 0 will be a real solution.
