#### Answer

(a) This is a quadratic equation.
(b) This is a not a quadratic equation.
(c) This is a quadratic equation.
(d) This is a quadratic equation.
(e) This is a quadratic equation for $x \geq \frac{1}{2}$
(f) This is a quadratic equation.

#### Work Step by Step

A quadratic equation can be written in this form:
$ax^2 + bx+c = 0$
where $a,b,$ and $c$ are real numbers and $a \neq 0$
(a) $2x^2-5x+3 = 0$
$a=2, b = -5, c=3$
This is a quadratic equation.
(b) $x^2 = x^2+4$
$-4 = 0$
$a=0, b = 0, c=-4$
This is a not a quadratic equation since $a=0$.
(c) $x^2 = 4$
$x^2-4 = 0$
$a=1, b = 0, c=-4$
This is a quadratic equation.
(d) $\frac{1}{2}x^2-\frac{1}{4}x-\frac{1}{8} = 0$
$a=\frac{1}{2}, b = -\frac{1}{4}, c=-\frac{1}{8}$
This is a quadratic equation.
(e) $\sqrt{2x-1} = 3$
$2x-1 = 9$
$(2x-1)^2 = 81$
$4x^2-4x+1 = 81$
$4x^2-4x-80 = 0$
$a=4, b = -4, c=-80$
This is a quadratic equation for $x \geq \frac{1}{2}$
Since the original equation included the term $\sqrt{2x-1}$, we need to restrict the domain to $x \geq \frac{1}{2}$
(f) $(x+1)(x-1) = 15$
$x^2-1 = 15$
$x^2-16 = 0$
$a=1, b = 0, c=-16$
This is a quadratic equation.