Let's solve the equation:
[tex]\[
3x^3 = 7x^2 - 4
\][/tex]
First, we rearrange the equation to set it equal to zero:
[tex]\[
3x^3 - 7x^2 + 4 = 0
\][/tex]
Now, we solve for [tex]\( x \)[/tex] by finding the roots of the polynomial [tex]\( 3x^3 - 7x^2 + 4 \)[/tex].
Let's consider potential rational roots of the polynomial. Rational roots, if they exist, would be of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term (4) and [tex]\( q \)[/tex] is a factor of the leading coefficient (3). The factors of 4 are [tex]\( \pm 1, \pm 2, \pm 4 \)[/tex], and the factors of 3 are [tex]\( \pm 1, \pm 3 \)[/tex]. Potential rational roots are thus:
[tex]\[
\pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}
\][/tex]
By evaluating the polynomial [tex]\( 3x^3 - 7x^2 + 4 \)[/tex] at these potential roots, we identify that:
1. [tex]\( x = -\frac{2}{3} \)[/tex]
2. [tex]\( x = 1 \)[/tex]
3. [tex]\( x = 2 \)[/tex]
are indeed the roots of the polynomial.
Therefore, the solutions to the equation [tex]\( 3x^3 = 7x^2 - 4 \)[/tex] are
[tex]\[
x = -\frac{2}{3}, \quad x = 1, \quad x = 2
\][/tex]
