How do you solve $x^4+36>=13x^2$ using a sign chart?

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How do you solve $x^4+36>=13x^2$ using a sign chart?

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Answer 1 · 2016-12-24T19:09:37

The answer is $x in ] -oo,-3 ] uu [ -2,2 ]uu [3, oo[$

Explanation:

We need

$a^2-b^2=(a+b)(a-b)$

Let's factorise the expression

$x^4-13x^2+36= (x^2-4)(x^2-9)$

$=(x+2)(x-2)(x+3)(x-3)$

Let $f(x)=(x+2)(x-2)(x+3)(x-3)$

We can now do the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-3$ $color(white)(aaaa)$ $-2$ $color(white)(aaaa)$ $2$ $color(white)(aaaa)$ $3$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x+3$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $x-2$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $x-3$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaa)$ $+$

Therefore,

$f(x>=0)$ , when $x in ] -oo,-3 ] uu [ -2,2 ]uu [3, oo[$

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How do you solve $x^4+36>=13x^2$ using a sign chart?

1 Answer

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How do you solve $x^4+36>=13x^2$ using a sign chart?