Question

$\boxed{ \boxed{ \pink{ \underbrace{ \huge{ \bf{QUESTION}}}}}}$
The Equation ￫


$\bold{ \sf{ \frac{ {24x}^{2} + 25x - 47 }{ax - 2} = - 8x - 3 - \frac{53}{ax - 2}}} \\ \\ \: is \: \: true \: for \: all \: values \: of \: x ≠\frac{2}{a} ,where \: a \: \: is \: constant \:$



What is the value of a?

A) -16

B) -3

C) 3

D) 16
​

Answer 1

EXPLANATION.

Equation :

⇒ 24x² + 25x - 47/ax - 2 = -8x - 3 - 53/ax - 2.

⇒ 24x² + 25x - 47/ax - 2 = (-8x - 3)(ax - 2) - 53/ax - 2.

Both (ax - 2) will get cancel,

⇒ 24x² + 25x - 47 = (-8x - 3)(ax - 2) - 53.

⇒ 24x² + 25x - 47 = (-8ax² + 16x - 3ax + 6) - 53.

⇒ 24x² + 25x - 47 = -8ax² + 16x - 3ax + 6 - 53.

⇒ 24x² + 25x - 47 = -8ax² - 3ax + 16x - 47.

⇒ 24x² + 8ax² + 3ax + 25x - 16x = 0.

⇒ (24 + 8a)x² + 3ax + 9x = 0.

⇒ (24 + 8a)x² + 3x( a + 3) = 0.

⇒ ( 3 + a)8x² + 3x( a + 3) = 0.

⇒ ( a + 3)( 8x² + 3x) = 0.

⇒ a ≠ 2/a.

⇒ a = -3.

Value of a = -3.

Option [ B] is correct answer.

                                                                                           

MORE INFORMATION.

Nature of roots,

The term b² - 4ac is called discriminant of the equation. it is denoted by Δ or D.

(A) = Suppose a, b, c ∈ R and a ≠ 0 then,

(1) = if D > 0 ⇒ Roots are real and unequal.

(2) = if D = 0 ⇒ Roots are real and equal and each equal to -b/2a.

(3) = if D < 0 ⇒ Roots are imaginary and unequal or complex conjugate.

(B) = Suppose a, b, c ∈ Q , a ≠ 0 then,

(1) = if D > 0 and D is perfect square ⇒ Roots are unequal and rational.

(2) = if D > 0 and D is not perfect square ⇒ Roots are irrational and unequal.

Answer 2

$\large\underline{\red{\sf \orange{\bigstar} Correct\;Question}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\bold{ \sf{ \frac{ {24x}^{2} + 25x - 47 }{ax - 2} = - 8x - 3 - \frac{53}{ax - 2}}} \\ \\ \: is \: \: true \: for \: all \: values \: of \: x ≠\frac{2}{a} ,where \: a \: \: is \: constant \:$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\large\underline{\pink{\sf \green{\bigstar} Solution}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\large\underline{\purple{\sf \orange{\bigstar} .}}$Equation $\large\underline{\red{\sf \blue{\bigstar} .}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 25x - 47/ax - 2 = -8x - 3 - 53/ax - 2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 25x - 47/ax - 2 = (-8x - 3)(ax - 2) - 53/ax - 2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\green{\sf{\star\; ☯\; Both\; (ax - 2) \;will \;get \;cancel \;☯}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 25x - 47 = (-8x - 3)(ax - 2) - 53.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 25x - 47 = (-8ax² + 16x - 3ax + 6) - 53.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 25x - 47 = -8ax² + 16x - 3ax + 6 - 53.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 25x - 47 = -8ax² - 3ax + 16x - 47.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ 24x² + 8ax² + 3ax + 25x - 16x = 0.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ (24 + 8a)x² + 3ax + 9x = 0.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ (24 + 8a)x² + 3x( a + 3) = 0.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ ( 3 + a)8x² + 3x( a + 3) = 0.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ ( a + 3)( 8x² + 3x) = 0

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ a ≠ 2/a.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ⇒ a = -3.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\pink{\sf{\star\;Value \;of \;a \;= \;-3.}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\orange{\sf{\star\;Option\; ( B) \;is \;correct \;answer.}}$

 ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀