If a quadratic equation has only one real solution, then can it or can it not be factored? Why?

Question

The quadratic equations contain integer coefficients.

Answer 1

If it has only one real solution, then you have a perfect square.  Suppose the equation is ax² + bx + c = 0 with only one solution x₁.  Then

 

a(x - x₁)² = 0

 

It is factored in the form

 

ax² + bx + c = a(x - x₁)(x - x₂) where x₁ = x₂ in this case.

 

Here is an example:

 

2x² + 4x + 2 = 2(x² + 2x + 1) = 2(x + 1)(x + 1) = 2(x + 1)², with one real solution x = -1

 

Does this help?

 

 

 

 

Answer 2

It can be factored.  You would have a quadratic equation such that

 

x² ± 2ax + a² = 0

 

 

Then this can be factored as

 

(x ± a)(x ± a) = 0

 

(x ± a)² = 0

 

 

where a is the integer.  The solution repeats.  Thus, it has a multiplicity of 2.