Find the real parameter a so that the equation has real and positive roots

Question

My problem is:

$(2-x)(x+1)=a$

How do I find the value of the parameter $a$ so that the equation has real and positive roots?

Answer 1

Your equation can be written

$$x^2-x+(a-2)=0$$

This is a quadratic equation, and it has real roots if it's discriminant $\Delta$ is $\geq0$, with $\Delta = 1-4\cdot(a-2)=9-4a$, so you need $a\leq 9/4=2{,}25$.

Then the roots are positive if both the sum of the roots and their product are positive. The sum is always $1$, from Vieta's formulas (see also here), and the product of the root is $a-2$, so you need $a>2$.

So, the condition is $2<a\leq 2{,}25$.