Question

${x}^{4} + {x}^{2} - 1 = 0$
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Answer 1

Answer:

Step-by-step explanation:

x^4 + x^2 - 1 = 0

x^4 + x^2 = 1

x doesnt exist since any number multiplied by itself even number of times (like 2^2, 2^4, 2^6 and so on), will always be positive.

since only 1/2 added twice gives 1, x doesnt exist

Answer 2

Given :

${x}^{4} + {x}^{2} - 1 = 0$

To Prove :

x = ?

Formula needed :

$= = > \: \: \: \: \: {b}^{2} - 4ac \\ \\ x \: = \frac{ - b \frac{ + }{ - } \sqrt{ {b}^{2} - 4ac} }{2a}$

Solution :

$write \: \: \: \: </strong><strong>{</strong><strong>(</strong><strong>x</strong><strong>)</strong><strong> </strong><strong>}^{4} </strong><strong> </strong><strong>\</strong><strong>:</strong><strong> </strong><strong>\</strong><strong>:</strong><strong> </strong><strong>\</strong><strong>:</strong><strong> </strong><strong>\</strong><strong>:</strong><strong> </strong><strong> </strong><strong> </strong><strong> in \: \: the \: \: form \: \: of \: \: { </strong><strong>(</strong><strong>{x}^{2}</strong><strong>)</strong><strong> </strong><strong>}^{2}$

Therefore ,,,, we get ,,,,,

${</strong><strong>(</strong><strong> {x}^{2} </strong><strong>)</strong><strong>}^{2} + {x}^{2} - 1 = 0$

This is in the form ,,,

$a {x}^{2} + bx + c = 0 \\ \\ where \: \: \: \: \: \: \: \: \\ x = {x}^{2} \\ a = 1 \\ b = 1 \\ c = </strong><strong>-</strong><strong> </strong><strong>1$

$</strong><strong>\</strong><strong>\</strong><strong> </strong><strong>\</strong><strong>\</strong><strong> </strong><strong>\</strong><strong>\</strong><strong> </strong><strong>= {b}^{2} - 4ac \\ \\ = {(1)}^{2} - 4 \times 1 \times - 1 \\ \\ = 1 + 4 \\ \\</strong><strong> </strong><strong>=</strong><strong> 5$

$x = \frac{ - b \frac{ + }{ - } \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ {x}^{2} = \frac{ - 1 \frac{ + }{ - } \sqrt{5} }{2 \times 1} \\ \\ {x}^{2} = \frac{ - 1 \frac{ + }{ - } \sqrt{5} }{2} \\ \\ x = \sqrt{ \frac{ - 1 \frac{ + }{ - } \sqrt{5} }{2} }$

where ,,,,

$x = \sqrt{ \frac{ - 1 + \sqrt{5} }{2} } \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: and \: \: \: \: \: \: \\ \\ \\ x = \sqrt{ \frac{ - 1 - \sqrt{</strong><strong>5</strong><strong>} }{2} }$

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