Question

A quadratic equation is shown below: 25x2 + 10x + 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 − 4x + 1 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used. (5 points)

Answer 1

A) The quadratic equation is:
$25x^2 + 10 x +1 =0$
which is of the form
$ax^2 + bx+c=0$
Let's analyze the radicand:
$b^2-4ac=(10)^2 - 4\cdot 25 \cdot 1=100-100 =0$
The radicand is zero: this means that the equation has 2 coincident solutions.

We can find them by using the formula:
$x= \frac{-b \pm \sqrt{b^2-4ac} }{2a}= \frac{-10 \pm 0}{2 \cdot 25}=-0.2$
So, the solution is x=-0.2 with multiplicity 2.

b) The equation is
$4x^2-4x+1=0$
Note that the radicand is zero again, as before:
$b^2-4ac=4^2 - 4\cdot 4 \cdot 1=16-16=0$
So we have two coincident solutions as before. We can find them using the same formula:
$x= \frac{-b \pm \sqrt{b^2-4ac} }{2a}= \frac{-(-4) \pm 0 }{2 \cdot 4}= \frac{4}{8}=0.5$
So, the solution is x=0.5 with multiplicity 2.