Question

Solve it:–
$\bf{ \dfrac{x - 3}{x + 3} \: + \: \dfrac{x + 3}{x - 3} \: = \: 2\dfrac{1}{2} }$
No spammed answers!​

Answer 1

Step-by-step explanation:

$\tt \red{ \leadsto \quad \: \dfrac{x - 3}{x + 3} \: + \: \dfrac{x + 3}{x - 3} \: = \: 2\dfrac{1}{2}}$

$\tt \red{\leadsto \quad\dfrac{ x-3 }{ x+3 } + \dfrac{ x+3 }{ x-3 } = \dfrac{ 5 }{ 2 }}$

$\tt \pink{\leadsto \quad\left(2x-6\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=5\left(x-3\right)\left(x+3\right) }$

$\tt \blue{ \leadsto \quad2x^{2}-12x+18+2x^{2}+12x+18=5\left(x-3\right)\left(x+3\right) }$

$\tt \orange{ \leadsto \quad4x^{2}-12x+18+12x+18=\left(5x-15\right)\left(x+3\right) }$

$\tt \green{ \leadsto \quad4x^{2} \cancel{-12x}+18+ \cancel{12x}+18=\left(5x - 15\right)\left(x+3\right) }$

$\tt \blue{ \leadsto \quad4x^{2}+36=5x^{2}-45 }$

$\tt \red{ \leadsto \quad-x^{2}+36=-45 }$

$\tt \pink{ \leadsto \quad-x^{2}=-81 }$

$\tt \blue{ \leadsto \quad \: x^{2}=81 }$

$\tt \purple{ \leadsto \quad \: x = \pm9}$

Answer 2

Step-by-step explanation:

$\sf \red{\frac{x - 3}{x + 3} + \frac{x + 3}{x - 3} = 2 \frac{1}{2} } \\ \\ \implies \: \frac{(x - 3)^{2} + (x + 3)^{2} }{(x + 3)(x - 3)} = \frac{5}{2}$

$\sf \implies \: \frac{ {x}^{2} - 6x + 9 + {x}^{2} + 6x + 9 }{ {x}^{2} - 9} = 5/2$

$\sf \implies \: 2(2 {x}^{2} + 18) = 5( {x}^{2} - 9) \\ \sf \implies \: 4 {x}^{2} + 36 = 5x2 - 45 \\ \sf \implies {x}^{2} - 81 = 0 \\ \sf \implies \: {x}^{2} - {9}^{2} = 0$

$\sf \implies \: (x + 9)(x - 9) = 0 \\ \\ \sf \red{if \: x + 9 = 0 \: or \: x - 9 = 0} \\ \sf \red{then \: x \: = - 9 \: or \: x \: = 9}$

Formulas for Solving Quadratic Equations

1. The roots of the quadratic equation: x = (-b ± √D)/2a, where D = b2 – 4ac

2. Nature of roots:

• D > 0, roots are real and distinct (unequal)
• D = 0, roots are real and equal (coincident)
• D < 0, roots are imaginary and unequal

3. The roots (α + iβ), (α – iβ) are the conjugate pair of each other.

4. Sum and Product of roots: If α and β are the roots of a quadratic equation, then

• S = α+β= -b/a = coefficient of x/coefficient of x2
• P = αβ = c/a = constant term/coefficient of x2

5. Quadratic equation in the form of roots: x2 – (α+β)x + (αβ) = 0

6. The quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have;

• One common root if (b1c2 – b2c1)/(c1a2 – c2a1) = (c1a2 – c2a1)/(a1b2 – a2b1)
• Both roots common if a1/a2 = b1/b2 = c1/c2

7. In quadratic equation ax2 + bx + c = 0 or [(x + b/2a)2 – D/4a2]

• If a > 0, minimum value = 4ac – b2/4a at x = -b/2a.
• If a < 0, maximum value 4ac – b2/4a at x= -b/2a.

8. If α, β, γ are roots of cubic equation ax3 + bx2 + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a