Question

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★ STANDARD QUESTION 78 ★

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Question is -

Determine the limits between which " n " must lie in order that the equation

2ax( ax + nc ) + ( n² - 2 )c² = 0
may have real roots .


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Answer 1

$given \: eqn \: \\ 2ax(ax + nc) + ( {n}^{2} - 2) {c}^{2} = 0 \\2 {a}^{2} {x}^{2} + 2ancx + ( {n}^{2} - 2 ) {c}^{2} = 0\\ to \: get \: real \: roots \: {b}^{2} - 4ac > = 0 \\ {(2anc)}^{2} - 4 \times 2 {a }^{2} \times ( {n}^{2} - 2) {c}^{2} > = 0 \\ 4 {a}^{2} {n}^{2} {c}^{2} - 8 {a}^{2} {c}^{2} ( {n}^{2} - 2) > = 0 \\ 4 {n}^{2} - 8( {n}^{2} - 2) > = 0 \\ - 4 {n}^{2} + 16 > = 0 \\ \: {n}^{2} < = 4 \\ - 4 {n}^{2} + 16 > = 0 \: only \: when \\ n > = - 2 \: and \: n < = 2 \\ so \: n \: belongs \: to \: ( - 2 \: 2) \: including \: - 2 \: and \: 2$
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