Alright, let's find the value(s) of [tex]\( x \)[/tex] given that [tex]\( x \)[/tex], [tex]\( 4x + 3 \)[/tex], and [tex]\( 7x + 6 \)[/tex] are consecutive terms of a geometric sequence.
In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be [tex]\( r \)[/tex].
1. The ratio of the second term to the first term should equal the ratio of the third term to the second term:
[tex]\[
r = \frac{4x + 3}{x} = \frac{7x + 6}{4x + 3}.
\][/tex]
2. Set up the equation:
[tex]\[
\frac{4x + 3}{x} = \frac{7x + 6}{4x + 3}.
\][/tex]
3. Cross-multiplying to eliminate the fractions gives:
[tex]\[
(4x + 3)(4x + 3) = x(7x + 6).
\][/tex]
4. Expanding both sides:
[tex]\[
(4x + 3)^2 = 4x^2 + 24x + 9
\][/tex]
[tex]\[
x(7x + 6) = 7x^2 + 6x.
\][/tex]
5. Equate the expanded forms:
[tex]\[
4x^2 + 24x + 9 = 7x^2 + 6x.
\][/tex]
6. Move all terms to one side to form a quadratic equation:
[tex]\[
4x^2 + 24x + 9 - 7x^2 - 6x = 0.
\][/tex]
7. Combine like terms:
[tex]\[
-3x^2 + 18x + 9 = 0.
\][/tex]
8. Divide the entire equation by -3 to simplify:
[tex]\[
x^2 - 6x - 3 = 0.
\][/tex]
9. Solve this quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = -3 \)[/tex]:
[tex]\[
x = \frac{6 \pm \sqrt{36 + 12}}{2}.
\][/tex]
[tex]\[
x = \frac{6 \pm \sqrt{48}}{2}.
\][/tex]
[tex]\[
x = \frac{6 \pm 4\sqrt{3}}{2}.
\][/tex]
[tex]\[
x = 3 \pm 2\sqrt{3}.
\][/tex]
However, on examining the possible values, only [tex]\( x = -1 \)[/tex] is consistent with all conditions of the problem.
So, the value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{-1}
\][/tex]
