Question

determine the discriminant of the quadratic equation. use the value of the discriminant to determine the number of solution(s) the quadratic equation has.

(x^2 - 8x + 9 = 0)

discriminant = 28

determine the number of solution(s) the quadratic equation has.

a. two complex solutions that are not real
b. one repeated real solution
c. two irrational solutions
d. two rational solutions

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Identify coefficients

For the quadratic equation \(x^2 - 8x + 9 = 0\):

$$a = 1, \quad b = -8, \quad c = 9$$

Step 2: Calculate the discriminant

The discriminant formula is \(D = b^2 - 4ac\):

$$D = (-8)^2 - 4(1)(9)$$

$$D = 64 - 36$$

$$D = 28$$

Step 3: Determine the nature of the solutions

Since the discriminant \(D = 28\) is:

• Positive (\(D > 0\)): There are two distinct real solutions.
• Not a perfect square: The square root of the discriminant, \(\sqrt{28}\), is irrational, which means the solutions will be irrational.

Therefore, the equation has two irrational solutions.

Answer:

• discriminant = \(28\)
• Number of solution(s): C. two irrational solutions