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How do you solve $\sqrt{5 x^{2} - 9} = 2 x$ $sqrt(5x^2-9)=2x$ and check your solution?

1 Answer

Sep 4, 2017

$x = + 3$ $x=+3$
(see below for solution and check)

Explanation:

Given
$XXX \sqrt{5 x^{2} - 9} = 2 x$ $color(white)("XXX")sqrt(5x^2-9)=2x$

Squaring both sides (this may introduce extraneous solutions; we will need to check later)
$XXX 5 x^{2} - 9 = 4 x^{2}$ $color(white)("XXX")5x^2-9=4x^2$

Subtract $4 x^{2}$ $4x^2$ from both sides
$XXX x^{2} - 9 = 0$ $color(white)("XXX")x^2-9=0$

Factor the left side
$XXX (x + 3) (x - 3) = 0$ $color(white)("XXX")(x+3)(x-3)=0$

which implies
$XXX x = - 3 xxx or xxx x = + 3$ $color(white)("XXX")x=-3color(white)("xxx") or color(white)("xxx")x=+3$

Checking for extraneous solutions:
${: ("with "x=-3,,color(white)("xxx"),"with "x=+3,), (sqrt(5 * x^2-9)=+6,2x=-6,,sqrt(5^2-9)=+6,2x=6), ("extraneous",,,"valid",) :}$

The only valid solution is $x=+3$

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