To solve the equation [tex]\( 7x^4 - 42x^2 = 35x \)[/tex] for all values of [tex]\( x \)[/tex], we need to follow several steps. Here is a detailed, step-by-step solution:
1. Rewrite the equation:
The original equation is:
[tex]\[
7x^4 - 42x^2 = 35x
\][/tex]
First, we bring all terms to one side of the equation to set it to zero:
[tex]\[
7x^4 - 42x^2 - 35x = 0
\][/tex]
2. Factor out common terms:
Notice that each term on the left-hand side has a common factor. Factor out [tex]\( 7x \)[/tex]:
[tex]\[
7x(x^3 - 6x - 5) = 0
\][/tex]
3. Break into individual factors:
According to the zero-product property, if the product of multiple factors equals zero, at least one of those factors must be zero:
[tex]\[
7x = 0 \quad \text{or} \quad x^3 - 6x - 5 = 0
\][/tex]
Solve each factor individually.
4. Solve [tex]\( 7x = 0 \)[/tex]:
[tex]\[
7x = 0 \Rightarrow x = 0
\][/tex]
So, one solution is [tex]\( x = 0 \)[/tex].
5. Solve the cubic equation [tex]\( x^3 - 6x - 5 = 0 \)[/tex]:
This equation is not easy to factor by inspection, so we resort to solving it either by using numerical methods, plotting, or algebraic techniques. However, based on known results, the real and complex roots of the equation are:
[tex]\[
x = -1, \quad x = \frac{1}{2} - \frac{\sqrt{21}}{2}, \quad x = \frac{1}{2} + \frac{\sqrt{21}}{2}
\][/tex]
In conclusion, the solutions to the equation [tex]\( 7x^4 - 42x^2 = 35x \)[/tex] are:
[tex]\[
x = -1, \quad x = 0, \quad x = \frac{1}{2} - \frac{\sqrt{21}}{2}, \quad x = \frac{1}{2} + \frac{\sqrt{21}}{2}
\][/tex]
