If the quadratic polynomial $lmx^2 + lmx + ln$ is a perfect square then prove that $4l^2=mn$

Question

I am new to this forum and request guidance on a question. I have looked into past answers on related topic but could not find one. Grateful for any help.

Answer 1

A polynomial is a perfect square if it is of the form $a^2 + 2ab + b^2 = (a+b)^2$. We'll do the same operation on $lmx^2 + mnx + ln$ so we get, $(\sqrt{lm}x)^2 + 2 \times \sqrt{lm}x \times \sqrt{ln} + (\sqrt{ln})^2$. Computing this we get, $lmx^2 + 2 \times \sqrt{lm} \times \sqrt{ln} \times x + ln$. But, our polynomial is $lmx^2 + mnx + ln$ So, equating both the polynomials we get, $2 \times \sqrt{lm} \times \sqrt{ln} = mn$ On squaring we get, $4 \times lm \times ln = m^2n^2$ From this, we can cancel out $mn$ from both the sides, so we get $4l^2 = mn$. Hence, we have proved it