Question

solve for x, 2x +4/x=9
​

Answer 1

GIVEN :–

$\\ \: \: { \huge{.}} \: \: \: { \bold{2x + \dfrac{4}{x} = 9 }}$

TO FIND :–

• Value of 'x' = ?

SOLUTION :–

$\\ \: \: \implies \: \: { \bold{2x + \dfrac{4}{x} = 9 }}$

$\\ \: \: \implies \: \: { \bold{2x {}^{2} + 4 = 9x }}$

$\\ \: \: \implies \: \: { \bold{2x {}^{2} - 9x + 4 = 0 }}$

• Let's use formula –

$\\ \: \: \longrightarrow \: \large { \boxed{ \bold{x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} }}}$

• Here –

$\\ \: \: \: \: \: \: { \huge{.}} \: \: \: { \bold{ \: a = 2 }}$

$\\ \: \: \: \: \: \: { \huge{.}} \: \: \: { \bold{ \: b = -9 }}$

$\\ \: \: \: \: \: \: { \huge{.}} \: \: \: { \bold{ \: c = 4 }}$

• Now put the values –

$\\ \: \: \implies \: { \bold{x = \dfrac{ - ( - 9) \pm \sqrt{ {( - 9)}^{2} - 4(2)(4)} }{2 \times 2} }}$

$\\ \: \: \implies \: { \bold{x = \dfrac{ 9 \pm \sqrt{ 81 - 32} }{4} }}$

$\\ \: \: \implies \: { \bold{x = \dfrac{ 9 \pm \sqrt{ 49} }{4} }}$

$\\ \: \: \implies \: { \bold{x = \dfrac{ 9 \pm 7 }{4} }}$

▪︎ Take positive(+) sign :–

$\\ \: \: \implies \: { \bold{x = \dfrac{ 9 + 7 }{4} }}$

$\\ \: \: \implies \: { \bold{x = \cancel\dfrac{ 16 }{4} }}$

$\\ \: \: \implies \: \large { \boxed{ \bold{x = 4 }}}$

▪︎ Take negative(-) sign :–

$\\ \: \: \implies \: { \bold{x = \dfrac{ 9 - 7 }{4} }}$

$\\ \: \: \implies \: { \bold{x = \cancel \dfrac{2}{4} }}$

$\\ \: \: \implies \: \large { \boxed{ \bold{x = \dfrac{1}{2} }}}$

Hence , The value of 'x' = 4 , ½

Answer 2

$\large\bf\underline{Given:-}$

• 2x + 4 /x = 9

$\large\bf\underline {To \: find:-}$

• Value of x.

$\huge\bf\underline{Solution:-}$

• p(x) = 2x + 4/x = 8

$\rm \mapsto \: 2x + \frac{4}{x} = 9 \\ \\ \rm \mapsto \: \frac{2 {x}^{2} + 4}{x} = 9 \\ \\ \rm \mapsto \: 2 {x}^{2} + 4 = 9x \\ \\ \rm \mapsto \: 2 {x}^{2} + 4 - 9x = 0 \\ \\ \rm \mapsto \: 2 {x}^{2} - 9x + 4 = 0 \\ \\ \rm \mapsto \: 2 {x}^{2} - 8x - x + 4 \\ \\ \rm \mapsto \: 2x(x - 4) - 1(x - 4) \\ \\ \rm \mapsto \: (2x - 1)(x - 4) \\ \\ \bf\mapsto \: x = \frac{1}{2} \: or \: x = 4$

Hence value of x = 1/2 or x = 4

$\bf\underline {Verification:-}$

$\rm \mapsto \: 2x + \frac{4}{x} = 9 \\ \\ \rm \mapsto \: \frac{2 {x}^{2} + 4}{x} = 9 \\ \\ \rm \mapsto \: 2 {x}^{2} + 4 = 9x \\ \\ \rm \mapsto \: 2 {x}^{2} + 4 - 9x = 0 \\ \\ \rm \mapsto \: 2 {x}^{2} - 9x + 4 = 0......(i)$

from equation (i) we can say that this is a quadratic equation.

p(x) = 2x² - 9x + 4 = 0

a = 2

b = -9

c = 4

Let α and β are the zeroes of the given polynomial.

α = 1/2

β = 4

sum of zeroes = -b/a

➝ 1/2 + 4 = -(-9)/2

➝ 1+8/2 = 9/2

➝ 9/2 = 9/2

Product of zeroes = c/a

➝ 1/2 × 4 = 4/2

➝ 4 /2 = 4/2

➝ 2 = 2

LHS = RHS

hence Verified.