Question

The number of real solutions of $\sqrt{({3x}^2-7x-30)}-\sqrt{({2x}^2-7x-5)}=x-5$ is

• A. one
• B. two
• C. three
• D. None of these

Answer 1

Answer: (A)

one

rewriting as

$\sqrt{3x^2-7x-30}=x-5+\sqrt{2x^2-7x-5}$

after squaring we get

$3x^2-7x-30-(x-5{)}^2-2x^2+7x+5=2(x-5)\sqrt{2x^2-7x-5}$

simplifying

$10x-50=2(x-5)\sqrt{2x^2-7x-5}$

$5x-25=(x-5)\sqrt{2x^2-7x-5}$

squaring again

$-(2x+5)(x-6)(x-5{)}^2=0$

Hence we get $4$ real value of $x$.