Question

How many real solutions does each of the following equations have?

A) x^2 + 11x + 12 = 0 (One solution)
B) 3x^2 - 2x + 2 = 0 (Two solutions)
C) x^2 + 4x + 4 = 0 (One solution)
D) 2x^2 - 2x + 3 = 0 (Two solutions)

Answer 1

The number of real solutions in each quadratic equation depends on the discriminant. By analyzing the discriminant, we can determine the number of real solutions for each equation. In this case, two of the equations have one real solution, while the other two have no real solutions.

Step-by-step explanation:

According to the given equations:

A) x^2 + 11x + 12 = 0 (One solution)

B) 3x^2 - 2x + 2 = 0 (Two solutions)

C) x^2 + 4x + 4 = 0 (One solution)

D) 2x^2 - 2x + 3 = 0 (Two solutions)

The number of real solutions in each equation can be determined by analyzing the discriminant of the quadratic equations.

A quadratic equation ax^2+bx+c=0 has:

1. Two real solutions if the discriminant (b^2-4ac) is positive.
2. One real solution if the discriminant is equal to zero.
3. No real solutions if the discriminant is negative.

By using the discriminant formula, we can determine:

• A) Discriminant = (11^2) - 4(1)(12) = 121 - 48 = 73 (Positive, therefore one solution).
• B) Discriminant = (-2^2) - 4(3)(2) = 4 - 24 = -20 (Negative, therefore no real solutions).
• C) Discriminant = (4^2) - 4(1)(4) = 16 - 16 = 0 (Zero, therefore one solution).
• D) Discriminant = (-2^2) - 4(2)(3) = 4 - 24 = -20 (Negative, therefore no real solutions).

Therefore, the answers are:

A) One solution

B) No real solutions

C) One solution

D) No real solutions