The equation ${| x - 2 |}^{10 x^{2} - 1} = {| x - 2 |}^{3 x}$

two positive and two negative solutions

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one positive and three negative solutions

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three positive and one negative solutions

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three positive and one zero

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Solution

The correct option is D three positive and one negative solutions
This is possible only if
$(1) | x - 2 | = 1 \Rightarrow x - 2 = \pm 1$
$x = 3$ or $x = 1$
OR
$(2) 10 x^{2} - 1 = 3 x$
$10 x^{2} - 3 x - 1 = 0$
$10 x^{2} - 5 x + 2 x - 1$
$5 x (2 x - 1) + 1 (2 x - 1) = 0$
$(5 x + 1) (2 x - 1) = 0$
$x = \frac{- 1}{5}$ or $x = \frac{1}{2}$
$∴ x = \frac{1}{2}, 1, 3, \frac{- 1}{5}$

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The equation |x−2|10x2−1=|x−2|3x

The correct option is D three positive and one negative solutionsThis is possible only if(1)|x−2|=1⇒x−2=±1x=3 or x=1OR(2)10x2−1=3x10x2−3x−1=010x2−5x+2x−15x(2x−1)+1(2x−1)=0(5x+1)(2x−1)=0x=−15 or x=12∴x=12,1,3,−15

The equation ${| x - 2 |}^{10 x^{2} - 1} = {| x - 2 |}^{3 x}$