Question

The Quadratic Formula and the Discrim

Complete parts a-c for each quadratic equation.
a. Find the value of the discriminant.
b. Describe the number and type of roots.
c. Find the exact solutions by using the Quadratic Formula.
1. x2 - 8x + 16 = 0

Answer 1

The discriminant for the equation x^2 - 8x + 16 is 0, indicating one real, repeated root. Applying the Quadratic Formula, we find the exact solution to be x = 4.

Step-by-step explanation:

The student has asked to solve the quadratic equation x2 - 8x + 16 = 0 by finding the discriminant, describing the number and type of roots, and using the Quadratic Formula for exact solutions.

Part a: Finding the Discriminant

The discriminant of a quadratic equation in the form ax2 + bx + c = 0 is given by b2 - 4ac. For the equation x2 - 8x + 16 = 0, we have a = 1, b = -8, and c = 16. Thus, the discriminant is (-8)2 - 4(1)(16) = 64 - 64 = 0.

Part b: Describing Roots

Sine the discriminant is 0, this indicates that there is exactly one real, repeated root.

Part c: Using the Quadratic Formula

The Quadratic Formula is x = (-b ± √(b2 - 4ac)) / (2a). Substituting our values into the formula gives us x = (8 ± √(0)) / (2(1)) = 8/2 = 4. Therefore, the exact solution is x = 4.