Question

Solve the following equation by factorization

${\displaystyle \sqrt{x(x-7)}=3\sqrt{2}}$

Answer 1

${\displaystyle \sqrt{x(x-7)}=3\sqrt{2}}$,
Squaring both sides,
x(x - 7) = 9 x 2
⇒ x² - 7x = 18
⇒ x² - 7x - 18 = 0,
⇒ x² - 9x + 2x - 18 = 0
⇒ x(x - 9) + 2(x - 9) = 0
⇒ (x - 9) (x + 2) = 0
Either x - 9 = 0,
then x = 9
or 
x + 2 = 0,
then x = -2
∴ x = 9, -2
check
(i) If x = 9, then
L.H.S.
= ${\displaystyle \sqrt{x(x-7)}}$
= ${\displaystyle \sqrt{9(9-7)}}$
= ${\displaystyle \sqrt{9\times 2}}$
= ${\displaystyle \sqrt{18}}$
= ${\displaystyle \sqrt{9\times 2}}$
= ${\displaystyle 3\sqrt{2}}$
= R.H.S.
x = 9 is a root
(ii) If x = -2, then
L.H.S.
= ${\displaystyle \sqrt{x(x-7)}}$
= ${\displaystyle \sqrt{-2(-2-7)}}$
= ${\displaystyle \sqrt{-2\times -9}}$
= ${\displaystyle \sqrt{18}}$
= ${\displaystyle \sqrt{9\times 2}}$
= ${\displaystyle 3\sqrt{2}}$
= R.H.S.
∴ x = -2 is also its root
Hence x = 9, -2.