Question

A quadratic equation is shown below:

Answer 1

Part A)

The given equation is:

$25 x^{2} +10x+1=0$

The radicand or discriminant(d) of the equation will be:

$d=(10)^{2}-4(25)(1) \\ \\ d=100-100 \\ \\ d=0$

Since the discriminant is equal to 0, the given quadratic equation has only 1 root. In other words we can say the the given equation is a perfect square. 


Part B)

The given equation is:

$4 x^{2} -4x+1=0$

We can solve this expression by factorization. Factors of middle term are to be made in such a way that their product equals the product of first and third term and sum is equal to the middle term i.e. product should be 4x² and sum should be -4x.
So the two such terms are -2x and -2x. Using the factors and simplifying the equation by taking common we get:

$4 x^{2} -2x-2x+1=0 \\ \\ 2x(2x-1)-1(2x-1)=0 \\ \\ (2x-1)(2x-1)=0 \\ \\ (2x-1) ^{2}=0 \\ \\ 2x-1=0 \\ \\ x= \frac{1}{2}$