Quadratic Equations - One rational solution?

Question

I have a question that I am working on:

Which of the following will give one rational solution?

4x^2 = 9
4x^2 - 12x = -9
x^2 = 5
x^2 - 2x + 14 = 0
2x^2 = x

I am not asking for the answer, I am actually trying to understand what the question means. i.e. What does it mean when it says "one rational solution"?

Answer 1

For example $$4x^2 = 9 \implies x=\pm\frac{3}{2}\tag{rational solutions}$$

and

$$x^2 = 5\implies x=\sqrt5\tag{irrational solutions}$$

for a general quadratic equation of the form $ax^2+bx+c=0$

by using quadratic formula we get, $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Now notice that for roots to be rational $b^2-4ac$ must be a perfect square and to have only one rational solution to $ax^2+bx+c=0$ we must have

$$x=\frac{-b+\sqrt{b^2-4ac}}{2a}=\frac{-b-\sqrt{b^2-4ac}}{2a}$$

$$\implies\sqrt{b^2-4ac}=0\implies \Delta=0$$

Answer 2

A rational solution will be a solution that is a rational number.

That is, the number will be of the form $$\dfrac{p}{q}, \text{ where }p,q\in\mathbb{Z}, q\neq 0$$

Answer 3

A quadratic equation that has only one rational solution will be one where:

• The discriminant equates to 0. (the expression underneath the square root sign will equal 0) This means that the curve will not cross the x-axis in two places.
• The solution is a rational one