To determine if the equation [tex]\((3x - 2)(2x - 3) = (2x + 5)(2x - 1)\)[/tex] is a quadratic equation, let's expand both sides of the equation and then compare the degrees of the resulting polynomials.
First, we expand the left side of the equation:
[tex]\[
(3x - 2)(2x - 3)
\][/tex]
Expanding this using distributive properties (FOIL method):
[tex]\[
= (3x)(2x) + (3x)(-3) + (-2)(2x) + (-2)(-3)
\][/tex]
[tex]\[
= 6x^2 - 9x - 4x + 6
\][/tex]
Combining like terms, we get:
[tex]\[
= 6x^2 - 13x + 6
\][/tex]
Next, we expand the right side of the equation:
[tex]\[
(2x + 5)(2x - 1)
\][/tex]
Expanding this using distributive properties (FOIL method):
[tex]\[
= (2x)(2x) + (2x)(-1) + (5)(2x) + (5)(-1)
\][/tex]
[tex]\[
= 4x^2 - 2x + 10x - 5
\][/tex]
Combining like terms, we get:
[tex]\[
= 4x^2 + 8x - 5
\][/tex]
Now, let's check the degrees of the polynomials on both sides of the equation.
The left side polynomial is:
[tex]\[
6x^2 - 13x + 6
\][/tex]
The highest power of [tex]\(x\)[/tex] in this polynomial is [tex]\(x^2\)[/tex], so its degree is 2.
The right side polynomial is:
[tex]\[
4x^2 + 8x - 5
\][/tex]
The highest power of [tex]\(x\)[/tex] in this polynomial is [tex]\(x^2\)[/tex], so its degree is also 2.
Since both sides of the equation are polynomials of degree 2, we can conclude that the equation [tex]\((3x - 2)(2x - 3) = (2x + 5)(2x - 1)\)[/tex] represents a quadratic equation.
