2. Without graphing determine the number of zeros for each function. a) f(x)=x^2-2x-14 b) g(x)=-3x^2+0.06x+4 c) f(x)=(1)/(4)x^2-3x+9 d) f(v)=-v^2+2v-1 e) f(x)=(1)/(2)x^2-x+(5)/(2) f) g(y)=-6y^2+5y-1

## Step 1: Calculate the Discriminant<br />### The number of zeros of a quadratic function $f(x) = ax^2 + bx + c$ is determined by the discriminant, $b^2 - 4ac$. If the discriminant is positive, there are two distinct real zeros. If the discriminant is zero, there is one real zero (a repeated root). If the discriminant is negative, there are no real zeros.<br /><br />## Step 2: Calculate Discriminant for a) $f(x) = x^2 - 2x - 14$<br />### $a=1$, $b=-2$, $c=-14$. Discriminant $= (-2)^2 - 4(1)(-14) = 4 + 56 = 60 > 0$. Two real zeros.<br /><br />## Step 3: Calculate Discriminant for b) $g(x) = -3x^2 + 0.06x + 4$<br />### $a=-3$, $b=0.06$, $c=4$. Discriminant $= (0.06)^2 - 4(-3)(4) = 0.0036 + 48 = 48.0036 > 0$. Two real zeros.<br /><br />## Step 4: Calculate Discriminant for c) $f(x) = \frac{1}{4}x^2 - 3x + 9$<br />### $a=\frac{1}{4}$, $b=-3$, $c=9$. Discriminant $= (-3)^2 - 4(\frac{1}{4})(9) = 9 - 9 = 0$. One real zero.<br /><br />## Step 5: Calculate Discriminant for d) $f(v) = -v^2 + 2v - 1$<br />### $a=-1$, $b=2$, $c=-1$. Discriminant $= (2)^2 - 4(-1)(-1) = 4 - 4 = 0$. One real zero.<br /><br />## Step 6: Calculate Discriminant for e) $f(x) = \frac{1}{2}x^2 - x + \frac{5}{2}$<br />### $a=\frac{1}{2}$, $b=-1$, $c=\frac{5}{2}$. Discriminant $= (-1)^2 - 4(\frac{1}{2})(\frac{5}{2}) = 1 - 5 = -4 < 0$. No real zeros.<br /><br />## Step 7: Calculate Discriminant for f) $g(y) = -6y^2 + 5y - 1$<br />### $a=-6$, $b=5$, $c=-1$. Discriminant $= (5)^2 - 4(-6)(-1) = 25 - 24 = 1 > 0$. Two real zeros.