Question

find the discriminant of each quadratic equation then determine if there are one, two or no solutions

1. $9n^2 - 3n - 8 = -10$
2. $-2x^2 - 8x - 14 = -6$
3. $9m^2 + 6m + 6 = 5$
4. $4a^2 = 8a - 4$
5. $-9b^2 = -8b + 8$
6. $-x^2 - 9 = 6x$

Explanation:

Problem 7: \( 9n^2 - 3n - 8 = -10 \)

Step 1: Rewrite in standard form

Add 10 to both sides: \( 9n^2 - 3n + 2 = 0 \). Here, \( a = 9 \), \( b = -3 \), \( c = 2 \).

Step 2: Calculate discriminant

Discriminant formula: \( D = b^2 - 4ac \). Substitute values: \( D = (-3)^2 - 4(9)(2) = 9 - 72 = -63 \).

Step 3: Determine solutions

Since \( D < 0 \), no real solutions.

Problem 8: \( -2x^2 - 8x - 14 = -6 \)

Step 1: Rewrite in standard form

Add 6 to both sides: \( -2x^2 - 8x - 8 = 0 \). Multiply by -1: \( 2x^2 + 8x + 8 = 0 \), so \( a = 2 \), \( b = 8 \), \( c = 8 \).

Step 2: Calculate discriminant

\( D = 8^2 - 4(2)(8) = 64 - 64 = 0 \).

Step 3: Determine solutions

Since \( D = 0 \), one real solution.

Problem 9: \( 9m^2 + 6m + 6 = 5 \)

Step 1: Rewrite in standard form

Subtract 5: \( 9m^2 + 6m + 1 = 0 \). Here, \( a = 9 \), \( b = 6 \), \( c = 1 \).

Step 2: Calculate discriminant

\( D = 6^2 - 4(9)(1) = 36 - 36 = 0 \).

Step 3: Determine solutions

Since \( D = 0 \), one real solution.

Problem 10: \( 4a^2 = 8a - 4 \)

Answer:

s:

1. Discriminant: \(-63\), No real solutions.
2. Discriminant: \(0\), One real solution.
3. Discriminant: \(0\), One real solution.
4. Discriminant: \(0\), One real solution.
5. Discriminant: \(-224\), No real solutions.
6. Discriminant: \(0\), One real solution.