Discriminant-of-a-Quadratic-Equation-Workbook - Free Printable

Problem Overview:

The task is to find the discriminant of each quadratic equation listed in the worksheet. The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

where:
- \( a \) is the coefficient of \( x^2 \),
- \( b \) is the coefficient of \( x \),
- \( c \) is the constant term.

The discriminant helps determine the nature of the roots of the quadratic equation:
- If \( \Delta > 0 \), the equation has two distinct real roots.
- If \( \Delta = 0 \), the equation has one real root (a repeated root).
- If \( \Delta < 0 \), the equation has no real roots (two complex conjugate roots).

Solution Approach:

For each quadratic equation, we will:
1. Identify the coefficients \( a \), \( b \), and \( c \).
2. Substitute these values into the discriminant formula \( \Delta = b^2 - 4ac \).
3. Simplify the expression to find the value of the discriminant.

Step-by-Step Solutions:

#### 1) \( 2n^2 + 2n + 1 = 0 \)
- \( a = 2 \), \( b = 2 \), \( c = 1 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 2^2 - 4(2)(1) = 4 - 8 = -4
\]
- Discriminant: \(\boxed{-4}\)

#### 2) \( -6a^2 - 2a - 10 = 0 \)
- \( a = -6 \), \( b = -2 \), \( c = -10 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-2)^2 - 4(-6)(-10) = 4 - 240 = -236
\]
- Discriminant: \(\boxed{-236}\)

#### 3) \( 6x^2 + 6x - 10 = 0 \)
- \( a = 6 \), \( b = 6 \), \( c = -10 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 6^2 - 4(6)(-10) = 36 + 240 = 276
\]
- Discriminant: \(\boxed{276}\)

#### 4) \( -4x^2 - 9x + 6 = 0 \)
- \( a = -4 \), \( b = -9 \), \( c = 6 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-9)^2 - 4(-4)(6) = 81 + 96 = 177
\]
- Discriminant: \(\boxed{177}\)

#### 5) \( -5x^2 - x + 4 = 0 \)
- \( a = -5 \), \( b = -1 \), \( c = 4 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-1)^2 - 4(-5)(4) = 1 + 80 = 81
\]
- Discriminant: \(\boxed{81}\)

#### 6) \( 2r^2 + 9r + 9 = 0 \)
- \( a = 2 \), \( b = 9 \), \( c = 9 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 9^2 - 4(2)(9) = 81 - 72 = 9
\]
- Discriminant: \(\boxed{9}\)

#### 7) \( n^2 + 9n + 8 = 0 \)
- \( a = 1 \), \( b = 9 \), \( c = 8 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 9^2 - 4(1)(8) = 81 - 32 = 49
\]
- Discriminant: \(\boxed{49}\)

#### 8) \( -5v^2 + 7v + 8 = 0 \)
- \( a = -5 \), \( b = 7 \), \( c = 8 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 7^2 - 4(-5)(8) = 49 + 160 = 209
\]
- Discriminant: \(\boxed{209}\)

#### 9) \( -5x^2 - 7x + 7 = 0 \)
- \( a = -5 \), \( b = -7 \), \( c = 7 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-7)^2 - 4(-5)(7) = 49 + 140 = 189
\]
- Discriminant: \(\boxed{189}\)

#### 10) \( -5b^2 - 9b + 5 = 0 \)
- \( a = -5 \), \( b = -9 \), \( c = 5 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-9)^2 - 4(-5)(5) = 81 + 100 = 181
\]
- Discriminant: \(\boxed{181}\)

#### 11) \( -3x^2 + 10x - 5 = -3 \)
- Rewrite as: \( -3x^2 + 10x - 2 = 0 \)
- \( a = -3 \), \( b = 10 \), \( c = -2 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 10^2 - 4(-3)(-2) = 100 - 24 = 76
\]
- Discriminant: \(\boxed{76}\)

#### 12) \( -2m^2 + 10m - 3 = -2 \)
- Rewrite as: \( -2m^2 + 10m - 1 = 0 \)
- \( a = -2 \), \( b = 10 \), \( c = -1 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 10^2 - 4(-2)(-1) = 100 - 8 = 92
\]
- Discriminant: \(\boxed{92}\)

#### 13) \( 4n^2 - 8n - 7 = -2 \)
- Rewrite as: \( 4n^2 - 8n - 5 = 0 \)
- \( a = 4 \), \( b = -8 \), \( c = -5 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-8)^2 - 4(4)(-5) = 64 + 80 = 144
\]
- Discriminant: \(\boxed{144}\)

#### 14) \( -k^2 + 3k - 11 = -3 \)
- Rewrite as: \( -k^2 + 3k - 8 = 0 \)
- \( a = -1 \), \( b = 3 \), \( c = -8 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 3^2 - 4(-1)(-8) = 9 - 32 = -23
\]
- Discriminant: \(\boxed{-23}\)

#### 15) \( 9b^2 - 4b + 14 = 9 \)
- Rewrite as: \( 9b^2 - 4b + 5 = 0 \)
- \( a = 9 \), \( b = -4 \), \( c = 5 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-4)^2 - 4(9)(5) = 16 - 180 = -164
\]
- Discriminant: \(\boxed{-164}\)

#### 16) \( 6a^2 + 5a + 17 = 7 \)
- Rewrite as: \( 6a^2 + 5a + 10 = 0 \)
- \( a = 6 \), \( b = 5 \), \( c = 10 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 5^2 - 4(6)(10) = 25 - 240 = -215
\]
- Discriminant: \(\boxed{-215}\)

#### 17) \( 3n^2 + 7n - 2 = -3 \)
- Rewrite as: \( 3n^2 + 7n + 1 = 0 \)
- \( a = 3 \), \( b = 7 \), \( c = 1 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 7^2 - 4(3)(1) = 49 - 12 = 37
\]
- Discriminant: \(\boxed{37}\)

#### 18) \( -2x^2 + 2x - 11 = -2 \)
- Rewrite as: \( -2x^2 + 2x - 9 = 0 \)
- \( a = -2 \), \( b = 2 \), \( c = -9 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 2^2 - 4(-2)(-9) = 4 - 72 = -68
\]
- Discriminant: \(\boxed{-68}\)

#### 19) \( 4r^2 - r - 1 = -3 \)
- Rewrite as: \( 4r^2 - r + 2 = 0 \)
- \( a = 4 \), \( b = -1 \), \( c = 2 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-1)^2 - 4(4)(2) = 1 - 32 = -31
\]
- Discriminant: \(\boxed{-31}\)

#### 20) \( 5p^2 - 8p - 6 = -2 \)
- Rewrite as: \( 5p^2 - 8p - 4 = 0 \)
- \( a = 5 \), \( b = -8 \), \( c = -4 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-8)^2 - 4(5)(-4) = 64 + 80 = 144
\]
- Discriminant: \(\boxed{144}\)

#### 21) \( 3v^2 = -7v \)
- Rewrite as: \( 3v^2 + 7v = 0 \) or \( 3v^2 + 7v + 0 = 0 \)
- \( a = 3 \), \( b = 7 \), \( c = 0 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 7^2 - 4(3)(0) = 49 - 0 = 49
\]
- Discriminant: \(\boxed{49}\)

#### 22) \( x^2 = -6 \)
- Rewrite as: \( x^2 + 6 = 0 \) or \( x^2 + 0x + 6 = 0 \)
- \( a = 1 \), \( b = 0 \), \( c = 6 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 0^2 - 4(1)(6) = 0 - 24 = -24
\]
- Discriminant: \(\boxed{-24}\)

#### 23) \( 8a^2 + 3a = -3 \)
- Rewrite as: \( 8a^2 + 3a + 3 = 0 \)
- \( a = 8 \), \( b = 3 \), \( c = 3 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 3^2 - 4(8)(3) = 9 - 96 = -87
\]
- Discriminant: \(\boxed{-87}\)

#### 24) \( 6p^2 + 10p = -9 \)
- Rewrite as: \( 6p^2 + 10p + 9 = 0 \)
- \( a = 6 \), \( b = 10 \), \( c = 9 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 10^2 - 4(6)(9) = 100 - 216 = -116
\]
- Discriminant: \(\boxed{-116}\)

Final Answer:

\[
\boxed{
\begin{array}{ll}
1) & -4 \\
2) & -236 \\
3) & 276 \\
4) & 177 \\
5) & 81 \\
6) & 9 \\
7) & 49 \\
8) & 209 \\
9) & 189 \\
10) & 181 \\
11) & 76 \\
12) & 92 \\
13) & 144 \\
14) & -23 \\
15) & -164 \\
16) & -215 \\
17) & 37 \\
18) & -68 \\
19) & -31 \\
20) & 144 \\
21) & 49 \\
22) & -24 \\
23) & -87 \\
24) & -116 \\
\end{array}
}
\]