Equation with p and q and expression with them

[MM]

Let p and q be such numbers that [tex]4p+4q+1<0[/tex] . Prove that the equation [tex]x^2-2qx+p=0[/tex] has real solutions.

[shyamjayakannan]

For the equation to have real roots, the discriminant should be [tex]\ge0[/tex]. So we need to show that [tex]4q^2-4p\ge0[/tex].

Now, [tex]4p+4q+1\lt0\Rightarrow4q+1\lt-4p\Rightarrow4q^2+4q+1\lt4q^2-4p\Rightarrow\left(2q+1\right)^2\lt4q^2-4p[/tex]

Since [tex]4q^2-4p[/tex] is greater than a perfect square, it is also greater than or equal to 0. Hence, proved.