Answer:
- x=2/3, -1/2 ±i√3/2
- x = -1, 0, 1
Step-by-step explanation:
When in doubt about how to solve a problem, it often works well to follow the instructions.
Solutions to 3x³ +x² = 2 -x
The first attachment shows a graph of the two sides of the equation. The point of intersection is the only real solution: x = 2/3.
The equation is 3rd-degree, so we know it has three solutions. When the equation is written in standard form, it is ...
3x³ +x² +x -2 = 0
The given real solution corresponds to a factor of ...
x -2/3
Multiplying by 3, this factor becomes (3x -2). When that is factored out of the standard form equation, the result is a quadratic factor with complex solutions. (See the second attachment for the division process.)
3x³ +x² +x -2 = (3x -2)(x² +x +1) = 0
The solutions to x² +x +1 = 0 can be found using the quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-1\pm\sqrt{1^2-4(1)(1)}}{2(1)}=\dfrac{-1\pm i\sqrt{3}}{2}[/tex]
All of the solutions to 3x³ +x² = 2 -x are ...
[tex]\boxed{x\in\left\{\dfrac{2}{3},\ \dfrac{-1\pm i\sqrt{3}}{2}\right\}}[/tex]
X-intercepts of g(x)
The x-intercepts are where g(x) = 0. The graph in the third attachment shows these handily.
You can also find the x-intercepts by factoring.
g(x) = x³ -x = x(x² -1)
g(x) = x(x -1)(x +1)
Values of x that make these factors zero are x=-1, x=0, x=1.
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Additional comments
The coefficients of a quadratic in standard form are used in the quadratic formula. That standard form is ax²+bx+c = 0. Comparing this to the given equation, you see that a=1, b=1, c=1.
A product is zero when one or more of its factors is zero. This is the "zero product rule" and is used to find the x-intercepts (zeros, solutions) of a quadratic. The solutions will make one or more of the factors be zero. Of course, the factor (x-a) is zero when x=a.
The factoring of the difference of squares is ...
a² -b² = (a -b)(a +b)
