x^4>36x^2 — step-by-step solution | Explain MAths

x^4>36x^2

1) Let's solve in

ℝ

:

x^{4} > 36 x^{2}

x^{4} - 36 x^{2} > 0

For

x^{4} - 36 x^{2}

, we recognize a difference of two squares:

{(x^{2})}^{2} - {(6 x)}^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x^{2}

and

b = 6 x

You can factorize the expression:

x^{4} - 36 x^{2}

into:

(x^{2} - 6 x) \cdot (x^{2} + 6 x)

2) Let's solve in

ℝ

:

(x^{2} - 6 x) \cdot (x^{2} + 6 x) > 0

We study the sign of each factor of the product

⇥2.1) Let's study the sign of

x^{2} - 6 x

by widening the sign to ≥ and solving in

ℝ

:

x^{2} - 6 x \geq 0

⇥You can factorize this expression into:

x \cdot (x - 6)

⇥We study the sign of each factor of the product

⇥⇥The expression

x

is zero at

x = 0

, it is negative before and positive after

⇥⇥The expression

x - 6

is zero at

x = 6

, it is negative before and positive after

⇥2.2) Let's study the sign of

x^{2} + 6 x

by widening the sign to ≥ and solving in

ℝ

:

x^{2} + 6 x \geq 0

⇥You can write the expression in factored form:

x \cdot (x + 6)

⇥We study the sign of each factor of the product

⇥⇥The expression

x

is zero at

x = 0

, it is negative before and positive after

⇥⇥The expression

x + 6

is zero at

x = -6

, it is negative before and positive after

3) Let's draw up the sign table:

x	$-∞$		$-6$		$0$		$6$		$+∞$
$x$		-		-	0	+		+
$x - 6$		-		-		-	0	+
$x$		-		-	0	+		+
$x + 6$		-	0	+		+		+
$x^{4} - 36 x^{2}$		+	0	-	0	-	0	+

▶Solution found:

x \in (-∞, -6) \cup (6, +∞)