How would I prove an equation (x^2=4x+3) has no integral solution?

[CSstudent]

I've looked up the question several times, it's only given me an equation that has two unknowns (I.E x^2=5y+2)

The question is:

Prove that x^2=4x+3 has no integral solutions

Help? Thank you!

 

[tkhunny]

I've looked up the question several times, it's only given me an equation that has two unknowns (I.E x^2=5y+2)

The question is:

Prove that x^2=4x+3 has no integral solutions

Help? Thank you!

Have a look at the "rational root theorem".

 

[JeffM]

Or try the integer root theorem.

 

[JeffM]

Alternatively, you could use the Quadratic Formula:

\(\displaystyle x^2 = 4x + 3 \implies x^2 - 4x - 3 = 0 \implies\)

\(\displaystyle x = \dfrac{-(-\ 4) \pm \sqrt{(-\ 4)^2 - 4(1)(-\ 3)}}{2 * 1} = \dfrac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7}.\)

Now can you show that neither solution is an integer?