Math worksheet for solving quadratic equations using the discriminant.

Problem Overview:

The task involves solving quadratic equations using the discriminant. The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

The discriminant determines the nature of the roots of the quadratic equation:
- If \( D > 0 \), the equation has two distinct real solutions.
- If \( D = 0 \), the equation has one real solution (a repeated root).
- If \( D < 0 \), the equation has no real solutions (the roots are complex).

We will solve the problems step by step.

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Part 1: Write out the formula for the discriminant

The formula for the discriminant is:

\[
\boxed{D = b^2 - 4ac}
\]

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Part 2: Determine the number of solutions based on the discriminant values

Given the following values of the discriminant \( D \), we determine whether the quadratic equation has two solutions, one solution, or no real solution:

1. \( D = 4 \)
- Since \( D > 0 \), the equation has two distinct real solutions.

2. \( D = -2 \)
- Since \( D < 0 \), the equation has no real solutions.

3. \( D = 3 \)
- Since \( D > 0 \), the equation has two distinct real solutions.

4. \( D = \frac{1}{3} \)
- Since \( D > 0 \), the equation has two distinct real solutions.

5. \( D = 0 \)
- Since \( D = 0 \), the equation has one real solution.

6. \( D = -\frac{5}{4} \)
- Since \( D < 0 \), the equation has no real solutions.

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Part 3: Identify \( a \), \( b \), and \( c \) in the given equations, and find the discriminant

We will identify the coefficients \( a \), \( b \), and \( c \) for each quadratic equation and then compute the discriminant \( D = b^2 - 4ac \).

#### 8. \( x^2 + 2x - 6 = 0 \)
- \( a = 1 \), \( b = 2 \), \( c = -6 \)
- Discriminant:
\[
D = b^2 - 4ac = 2^2 - 4(1)(-6) = 4 + 24 = 28
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 9. \( 3x^2 - x - 4 = 12 \)
First, rewrite the equation in standard form:
\[
3x^2 - x - 4 - 12 = 0 \implies 3x^2 - x - 16 = 0
\]
- \( a = 3 \), \( b = -1 \), \( c = -16 \)
- Discriminant:
\[
D = b^2 - 4ac = (-1)^2 - 4(3)(-16) = 1 + 192 = 193
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 10. \( 4x^2 + 3x = 8 \)
Rewrite the equation in standard form:
\[
4x^2 + 3x - 8 = 0
\]
- \( a = 4 \), \( b = 3 \), \( c = -8 \)
- Discriminant:
\[
D = b^2 - 4ac = 3^2 - 4(4)(-8) = 9 + 128 = 137
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 11. \( \frac{2x}{3} - x^2 + \frac{1}{3} = 0 \)
Rewrite the equation in standard form:
\[
-x^2 + \frac{2x}{3} + \frac{1}{3} = 0
\]
Multiply through by 3 to clear the fractions:
\[
-3x^2 + 2x + 1 = 0
\]
- \( a = -3 \), \( b = 2 \), \( c = 1 \)
- Discriminant:
\[
D = b^2 - 4ac = 2^2 - 4(-3)(1) = 4 + 12 = 16
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 12. \( x^2 + 7x - 2 = 10 \)
Rewrite the equation in standard form:
\[
x^2 + 7x - 2 - 10 = 0 \implies x^2 + 7x - 12 = 0
\]
- \( a = 1 \), \( b = 7 \), \( c = -12 \)
- Discriminant:
\[
D = b^2 - 4ac = 7^2 - 4(1)(-12) = 49 + 48 = 97
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 13. \( 15 - 5x = 3x^2 + 4x \)
Rewrite the equation in standard form:
\[
15 - 5x - 3x^2 - 4x = 0 \implies -3x^2 - 9x + 15 = 0
\]
Multiply through by \(-1\) to simplify:
\[
3x^2 + 9x - 15 = 0
\]
- \( a = 3 \), \( b = 9 \), \( c = -15 \)
- Discriminant:
\[
D = b^2 - 4ac = 9^2 - 4(3)(-15) = 81 + 180 = 261
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 14. \( 19x - x^2 = 3x + 2x^2 - 5 \)
Rewrite the equation in standard form:
\[
19x - x^2 - 3x - 2x^2 + 5 = 0 \implies -3x^2 + 16x + 5 = 0
\]
Multiply through by \(-1\) to simplify:
\[
3x^2 - 16x - 5 = 0
\]
- \( a = 3 \), \( b = -16 \), \( c = -5 \)
- Discriminant:
\[
D = b^2 - 4ac = (-16)^2 - 4(3)(-5) = 256 + 60 = 316
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

#### 15. \( x^2 + 2x - 3 = 4 \)
Rewrite the equation in standard form:
\[
x^2 + 2x - 3 - 4 = 0 \implies x^2 + 2x - 7 = 0
\]
- \( a = 1 \), \( b = 2 \), \( c = -7 \)
- Discriminant:
\[
D = b^2 - 4ac = 2^2 - 4(1)(-7) = 4 + 28 = 32
\]
- Since \( D > 0 \), the equation has two distinct real solutions.

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Final Answers:

1. Discriminant formula: \( \boxed{D = b^2 - 4ac} \)
2. Number of solutions based on discriminant values:
- \( D = 4 \): Two solutions
- \( D = -2 \): No real solutions
- \( D = 3 \): Two solutions
- \( D = \frac{1}{3} \): Two solutions
- \( D = 0 \): One solution
- \( D = -\frac{5}{4} \): No real solutions
3. Discriminants and nature of solutions for the equations:
- 8: \( D = 28 \), Two solutions
- 9: \( D = 193 \), Two solutions
- 10: \( D = 137 \), Two solutions
- 11: \( D = 16 \), Two solutions
- 12: \( D = 97 \), Two solutions
- 13: \( D = 261 \), Two solutions
- 14: \( D = 316 \), Two solutions
- 15: \( D = 32 \), Two solutions

\boxed{\text{All equations have two distinct real solutions.}}