Question

A quadratic equation is shown below:

16x2 − 8x + 1 = 0

Part A: Describe the solution(s) to the equation by just determining the radicand. Show your work. (5 points)

Part B: Solve 4x2 − 8x − 45 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used. (5 points)

Answer 1

First problem
We want to solve 16x² - 8x + 1 = 0

Notice that the quadratic expression is a perfect square because
(4x - 1)² = (4x)² + 2(4x)(-1) + (-1)² = 16x² - 8x + 1
Therefore
(4x - 1)² = 0
4x - 1 = 0  => x = 1/4 (multiplicity of 2)

Answer: x = 1/4 (multiplicity of 2)

Second problem
We want to solve 4x² - 8x - 45 = 0

The discriminant is
D = √[(-8)² - 4(4)(-45)]
    = √784
    = 28
Because D>0, there will be two real roots.

Use the quadratic formula.
x = (1/8)[8 +/- 28]
   = (1/8)(36) = 4.5
or
x = (1/8)(-20) 
   = -2.5

Answer:  x = 4.5 or x = -2.5