The equation  \bf \dfrac{24 {x}^{2} + 25x - 47 }{ax-2} = - 8x - 3 - \dfrac{53}{ax - 2} is true for all values of [tex] \bf

Question

The equation  \bf  \dfrac{24 {x}^{2} + 25x - 47 }{ax-2}  =  - 8x - 3 -  \dfrac{53}{ax - 2} is true for all values of  \bf x \neq \dfrac{2}{a} , where a is a constant.

What is the value of a?

(A) -16

(B) -3

(C) 3

(D) 16​

in progress 0
Arianna 1 month 2021-08-13T08:56:39+00:00 2 Answers 1 views 0

Answers ( )

    0
    2021-08-13T08:57:51+00:00

    EXPLANATION.

    equation are

    =>

      \bold{\frac{24 {x}^{2}  + 25x - 47}{ax - 2}  =  - 8x - 3 -  \frac{53}{ax - 2}}

    conditions are = x ≠ 2/a , where a is

    constant.

    TO FIND VALUE OF A.

    multiply both sides of equation by ( ax – 2 )

    we get,

    => 24x² + 25x – 47 = ( ax – 2 ) ( -8x – 3 ) – 53

    => 24x² + 25x + 6 = ( ax – 2 ) ( -8x – 3 )

    => 24x² + 25x + 6 = – 8ax² – 3ax + 16x + 6

    => 24x² + 9x = -8ax² – 3ax

    => 24x² + 8ax² + 9x + 3ax = 0

    => 8x² ( 3 + a) + 3x ( 3 + a) = 0

    => ( 8x² + 3x ) ( a + 3 ) = 0

    => a = -3

    for the given conditions

    => x ≠ 2/a

    => x ≠ -2/3

    Therefore,

    value of a = -3

    0
    2021-08-13T08:58:25+00:00

    ♣ Qᴜᴇꜱᴛɪᴏɴ :

    The equation  \sf{\dfrac{24 \mathrm{x}^{2}+25 \mathrm{x}-47}{\mathrm{ax}-2}=-8 \mathrm{x}-3-\dfrac{53}{\mathrm{ax}-2}} is true for all values of \sf{x \neq \dfrac{2}{a}} , where a is a constant.

    ♣ ɢɪᴠᴇɴ :


    \sf{\dfrac{24 \mathrm{x}^{2}+25 \mathrm{x}-47}{\mathrm{ax}-2}=-8 \mathrm{x}-3-\dfrac{53}{\mathrm{ax}-2}}

    ♣ ᴛᴏ ꜰɪɴᴅ :

    Value of a


    ♣ ᴀɴꜱᴡᴇʀ :

    \large{\boxed{\sf{a=-3}}}

    ♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

    \boxed{\underline{\underline{\sf{\dfrac{24 \mathrm{x}^{2}+25 \mathrm{x}-47}{\mathrm{ax}-2}=-8 \mathrm{x}-3-\dfrac{53}{\mathrm{ax}-2}}}}}

    \sf{\Rightarrow \dfrac{24 \mathrm{x}^{2}+25 \mathrm{x}-47}{\mathrm{ax}-2}=\dfrac{(-8 \mathrm{x}-3)(\mathrm{ax}-2)-53}{{ax}-2}}

    \sf{\Rightarrow 24 \mathrm{x}^{2}+25 \mathrm{x}-47=(-8 \mathrm{x}-3)(\mathrm{ax}-2)-53}

    \sf{\Rightarrow 24 \mathrm{x}^{2}+25 \mathrm{x}-47=-8 \mathrm{ax}^{2}+16 \mathrm{x}-3 \mathrm{ax}+6-53}

    \sf{\Rightarrow 24 \mathrm{x}^{2}+25 \mathrm{x}-47=-8 \mathrm{ax}^{2}+16 \mathrm{x}-3 \mathrm{ax}-47}

    \sf{\Rightarrow 24 \mathrm{x}^{2}+25 \mathrm{x}=-8 \mathrm{ax}^{2}+(16-3 \mathrm{a}) \mathrm{x}}

    \sf{\Rightarrow 24=-8 \mathrm{a}}

    \large{\boxed{\bigstar\:\:\sf{a=-3}}}

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