Question

find the discriminant. then, state whether the equation has 1, 2, or no real number solutions.

1. $5x^2 + 50x + 125 = 0$
2. $x^2 = x + 3$

1. $2x^2 + 23 = 14x$
2. $4x^2 + 7x - 15 = 0$

1. $8x^2 + 6x + 5 = 0$
2. $3x^2 + 3 = 6x$

1. $4x^2 - x = 8$
2. $10x^2 - x + 9 = 0$

Explanation:

Problem 1: \( 5x^2 + 50x + 125 = 0 \)

Step 1: Identify \( a \), \( b \), \( c \)

For a quadratic equation \( ax^2 + bx + c = 0 \), here \( a = 5 \), \( b = 50 \), \( c = 125 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = 50^2 - 4 \times 5 \times 125 \)
\( = 2500 - 2500 \)
\( = 0 \)

Step 3: Determine the number of solutions

If \( D = 0 \), the equation has 1 real solution.

Step 1: Identify \( a \), \( b \), \( c \)

\( a = 1 \), \( b = -1 \), \( c = -3 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = (-1)^2 - 4 \times 1 \times (-3) \)
\( = 1 + 12 \)
\( = 13 \)

Step 3: Determine the number of solutions

Since \( D = 13 > 0 \), the equation has 2 real solutions.

Step 1: Identify \( a \), \( b \), \( c \)

\( a = 2 \), \( b = -14 \), \( c = 23 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = (-14)^2 - 4 \times 2 \times 23 \)
\( = 196 - 184 \)
\( = 12 \)

Step 3: Determine the number of solutions

Since \( D = 12 > 0 \), the equation has 2 real solutions.

Answer:

Discriminant: \( 0 \); Number of solutions: \( 1 \)

Problem 2: \( x^2 = x + 3 \) (rewrite as \( x^2 - x - 3 = 0 \))