Question

(a-1)x²+2x-3=0 has equal roots then the value of a is?

Answer 1

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm :\longmapsto\:(a - 1) {x}^{2} + 2x - 3 = 0$

On comparing with ax² + bx + c = 0, we get

$\red{\rm :\longmapsto\:a = a - 1}$

$\red{\rm :\longmapsto\:b = 2}$

$\red{\rm :\longmapsto\:c = - 3}$

We know, the quadratic equation ax² + bx + c = 0 have real and equal roots iff Discriminant, D = b² - 4ac = 0

Thus,

$\rm :\longmapsto\: {2}^{2} - 4(a - 1)( - 3) = 0$

$\rm :\longmapsto\: 4 + 12(a - 1)= 0$

$\rm :\longmapsto\: 4 + 12a - 12= 0$

$\rm :\longmapsto\: 12a - 8= 0$

$\rm :\longmapsto\: 12a = 8$

$\rm :\longmapsto\:a = \dfrac{8}{12}$

$\bf\implies \:a = \dfrac{2}{3}$

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Concept Used :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Answer 2

Step-by-step explanation:

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm :\longmapsto\:(a - 1) {x}^{2} + 2x - 3 = 0$

On comparing with ax² + bx + c = 0, we get

$\red{\rm :\longmapsto\:a = a - 1}$

$\red{\rm :\longmapsto\:b = 2}$

$\red{\rm :\longmapsto\:c = - 3}$

We know, the quadratic equation ax² + bx + c = 0 have real and equal roots iff Discriminant, D = b² - 4ac = 0

Thus,

$\rm :\longmapsto\: {2}^{2} - 4(a - 1)( - 3) = 0$

$\rm :\longmapsto\: 4 + 12(a - 1)= 0$

$\rm :\longmapsto\: 4 + 12a - 12= 0$

$\rm :\longmapsto\: 12a - 8= 0$

$\rm :\longmapsto\: 12a = 8$

$\rm :\longmapsto\:a = \dfrac{8}{12}$

$\bf\implies \:a = \dfrac{2}{3}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Concept Used :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac