Find the discriminant of each quadratic equation on this math worksheet.

To solve the problem of finding the discriminant of each quadratic equation, we will use the formula for the discriminant of a quadratic equation:

$$
\Delta = b^2 - 4ac
$$

where the quadratic equation is in the standard form:
$$
ax^2 + bx + c = 0
$$

Step-by-Step Solution:

#### 1) \( 2v^2 + 3v + 5 = 0 \)
- Here, \( a = 2 \), \( b = 3 \), and \( c = 5 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = 3^2 - 4(2)(5) = 9 - 40 = -31
$$
- Answer: \(\Delta = -31\)

#### 2) \( t^2 - 9t - 1 = 0 \)
- Here, \( a = 1 \), \( b = -9 \), and \( c = -1 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = (-9)^2 - 4(1)(-1) = 81 + 4 = 85
$$
- Answer: \(\Delta = 85\)

#### 3) \( 6x^2 + x - 2 = 0 \)
- Here, \( a = 6 \), \( b = 1 \), and \( c = -2 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = 1^2 - 4(6)(-2) = 1 + 48 = 49
$$
- Answer: \(\Delta = 49\)

#### 4) \( 3z^2 - 5z + 10 = 0 \)
- Here, \( a = 3 \), \( b = -5 \), and \( c = 10 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = (-5)^2 - 4(3)(10) = 25 - 120 = -95
$$
- Answer: \(\Delta = -95\)

#### 5) \( n^2 + 2n + 4 = 0 \)
- Here, \( a = 1 \), \( b = 2 \), and \( c = 4 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = 2^2 - 4(1)(4) = 4 - 16 = -12
$$
- Answer: \(\Delta = -12\)

#### 6) \( 10h^2 - h = 0 \)
- First, rewrite the equation in standard form: \( 10h^2 - h + 0 = 0 \).
- Here, \( a = 10 \), \( b = -1 \), and \( c = 0 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = (-1)^2 - 4(10)(0) = 1 - 0 = 1
$$
- Answer: \(\Delta = 1\)

#### 7) \( 4m^2 - 9 = 0 \)
- First, rewrite the equation in standard form: \( 4m^2 + 0m - 9 = 0 \).
- Here, \( a = 4 \), \( b = 0 \), and \( c = -9 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = 0^2 - 4(4)(-9) = 0 + 144 = 144
$$
- Answer: \(\Delta = 144\)

#### 8) \( g^2 - 7g + 1 = 0 \)
- Here, \( a = 1 \), \( b = -7 \), and \( c = 1 \).
- Discriminant:
$$
\Delta = b^2 - 4ac = (-7)^2 - 4(1)(1) = 49 - 4 = 45
$$
- Answer: \(\Delta = 45\)

Final Answers:

$$
\boxed{
\begin{array}{ll}
1) & -31 \\
2) & 85 \\
3) & 49 \\
4) & -95 \\
5) & -12 \\
6) & 1 \\
7) & 144 \\
8) & 45 \\
\end{array}
}
$$