Question

The equation px²-22x+8=0 has two distinct roots a and b
1/α-1/β=9/4
Find the value of p​

Answer 1

Question

The equation px²-22x+8=0 has two distinct roots a and b1/α-1/β=9/4Find the value of p

Answer:

5

solution

given

$\frac { 1 } { \alpha } - \frac { 1 } { B } = \frac { 9 } { 4 }$

$lcm \: of \: \alpha and \beta \: is \: \\ \\ \frac{ \beta - \alpha }{ \alpha \beta } = \frac{9}{4}$

$as \: we \: know \: sum \: of \: zeros \: \alpha + \beta = \frac{ - b}{a} = \frac{22}{p} \\ \\ and \: product \: of \: zeros = \alpha \beta = \frac{c}{a} = \frac{8}{p}$

there fore by using formula

$(B−α)2=(α+B)2−4αB \\ \\ </strong></p><p></p><p><strong>[tex](B−α)2=(α+B)2−4αB \\ \\ ( B - \alpha ) ^ { 2 } = ( \frac { 22 } { p } ) ^ { 2 } - \frac { 4 ( 8 ) } { P }</strong></p><p></p><p><strong>[tex](B−α)2=(α+B)2−4αB \\ \\ ( B - \alpha ) ^ { 2 } = ( \frac { 22 } { p } ) ^ { 2 } - \frac { 4 ( 8 ) } { P }$

on squring both sides

${ \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\</strong></p><p></p><p><strong>[tex] { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\so \: by \: putting \: values \\ </strong></p><p></p><p><strong>[tex] { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\so \: by \: putting \: values \\ \frac { 484 - 32 p } { 64 } = \frac { 81 } { 16 } \\ \\now \: by \: bcm \\ 16(484 - 32p) = 81 \times 64</strong></p><p></p><p><strong>[tex] { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\so \: by \: putting \: values \ \\ \\now \: by \: bcm \\ 16(484 - 32p) = 81 \times 64$

$7744 - 512p \: = 5,184 \\ \\ - 512p = 5184 - 7744 \\ \\ \\ - 512p = - 2560 \\ \\ then \: p \: = 5$

hope it helps

be brainly

Answer 2

$Question</p><p>The equation px²-22x+8=0 has two distinct roots a and b1/α-1/β=9/4Find the value of p</p><p></p><p>Answer:</p><p>5</p><p></p><p>solution</p><p>given</p><p></p><p>\frac { 1 } { \alpha } - \frac { 1 } { B } = \frac { 9 } { 4 } </p><p>α</p><p>1</p><p> </p><p> − </p><p>B</p><p>1</p><p> </p><p> = </p><p>4</p><p>9</p><p> </p><p> </p><p></p><p>\begin{gathered}lcm \: of \: \alpha and \beta \: is \: \\ \\ \frac{ \beta - \alpha }{ \alpha \beta } = \frac{9}{4} \end{gathered} </p><p>lcmofαandβis</p><p>αβ</p><p>β−α</p><p> </p><p> = </p><p>4</p><p>9</p><p> </p><p> </p><p> </p><p> </p><p></p><p>\begin{gathered}as \: we \: know \: sum \: of \: zeros \: \alpha + \beta = \frac{ - b}{a} = \frac{22}{p} \\ \\ and \: product \: of \: zeros = \alpha \beta = \frac{c}{a} = \frac{8}{p} \end{gathered} </p><p>asweknowsumofzerosα+β= </p><p>a</p><p>−b</p><p> </p><p> = </p><p>p</p><p>22</p><p> </p><p> </p><p>andproductofzeros=αβ= </p><p>a</p><p>c</p><p> </p><p> = </p><p>p</p><p>8</p><p> </p><p> </p><p> </p><p> </p><p></p><p>there fore by using formula</p><p></p><p>\begin{gathered}(B−α)2=(α+B)2−4αB \\ \\ < /strong > < /p > < p > < /p > < p > < strong > [tex](B−α)2=(α+B)2−4αB \\ \\ ( B - \alpha ) ^ { 2 } = ( \frac { 22 } { p } ) ^ { 2 } - \frac { 4 ( 8 ) } { P } < /strong > < /p > < p > < /p > < p > < strong > [tex](B−α)2=(α+B)2−4αB \\ \\ ( B - \alpha ) ^ { 2 } = ( \frac { 22 } { p } ) ^ { 2 } - \frac { 4 ( 8 ) } { P }\end{gathered} </p><p>(B−α)2=(α+B)2−4αB</p><p></strong></p><p></p><p><strong>[tex](B−α)2=(α+B)2−4αB</p><p>(B−α) </p><p>2</p><p> =( </p><p>p</p><p>22</p><p> </p><p> ) </p><p>2</p><p> − </p><p>P</p><p>4(8)</p><p> </p><p> </strong></p><p></p><p><strong>[tex](B−α)2=(α+B)2−4αB</p><p>(B−α) </p><p>2</p><p> =( </p><p>p</p><p>22</p><p> </p><p> ) </p><p>2</p><p> − </p><p>P</p><p>4(8)</p><p> </p><p> </p><p> </p><p> </p><p></p><p>on squring both sides</p><p></p><p>\begin{gathered} { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\ < /strong > < /p > < p > < /p > < p > < strong > [tex] { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\so \: by \: putting \: values \\ < /strong > < /p > < p > < /p > < p > < strong > [tex] { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\so \: by \: putting \: values \\ \frac { 484 - 32 p } { 64 } = \frac { 81 } { 16 } \\ \\now \: by \: bcm \\ 16(484 - 32p) = 81 \times 64 < /strong > < /p > < p > < /p > < p > < strong > [tex] { \frac{ \beta - \alpha }{ \alpha \beta } }^{2} = { \frac{9}{4} }^{2} \\ \\so \: by \: putting \: values \ \\ \\now \: by \: bcm \\ 16(484 - 32p) = 81 \times 64\end{gathered} </p><p>αβ</p><p>β−α</p><p> </p><p> </p><p>2</p><p> = </p><p>4</p><p>9</p><p> </p><p> </p><p>2</p><p> </p><p></strong></p><p></p><p><strong>[tex] </p><p>αβ</p><p>β−α</p><p> </p><p> </p><p>2</p><p> = </p><p>4</p><p>9</p><p> </p><p> </p><p>2</p><p> </p><p>sobyputtingvalues</p><p></strong></p><p></p><p><strong>[tex] </p><p>αβ</p><p>β−α</p><p> </p><p> </p><p>2</p><p> = </p><p>4</p><p>9</p><p> </p><p> </p><p>2</p><p> </p><p>sobyputtingvalues</p><p>64</p><p>484−32p</p><p> </p><p> = </p><p>16</p><p>81</p><p> </p><p> </p><p>nowbybcm</p><p>16(484−32p)=81×64</strong></p><p></p><p><strong>[tex] </p><p>αβ</p><p>β−α</p><p> </p><p> </p><p>2</p><p> = </p><p>4</p><p>9</p><p> </p><p> </p><p>2</p><p> </p><p>sobyputtingvalues </p><p>nowbybcm</p><p>16(484−32p)=81×64</p><p> </p><p> </p><p></p><p>\begin{gathered}7744 - 512p \: = 5,184 \\ \\ - 512p = 5184 - 7744 \\ \\ \\ - 512p = - 2560 \\ \\ then \: p \: = 5\end{gathered} </p><p>7744−512p=5,184</p><p>−512p=5184−7744</p><p>−512p=−2560</p><p>thenp=5</p><p> </p><p> </p><p></p><p>hope it helps</p><p></p><p>be brainly$