Question

Determine k such that the quadratic equation
${x}^{2} + 7(3 + 2k) – 2x (1 + 3k) = 0$
has equal roots. ​

Answer 1

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

Given quadratic equation is

$\rm :\longmapsto\: {x}^{2} - 2x(1 + 3k) + 7(3 + 2k) = 0$

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

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

$\red{\rm :\longmapsto\:b = - 2(1 + 3k)}$

$\red{\rm :\longmapsto\:c = 7(3 + 2k)}$

We know,

Quadratic equation ax² + bx + c = 0, have real and equal roots iff Discriminant, D = 0.

$\rm :\longmapsto\: {b}^{2} - 4ac = 0$

$\rm :\longmapsto\: {\bigg[ - 2(1 + 3k)\bigg]}^{2} - 4(1)[7(3 + 2k] = 0$

$\rm :\longmapsto\:4 {(1 + 3k)}^{2} - 28(3 + 2k) = 0$

$\rm :\longmapsto\:4 \bigg[{(1 + 3k)}^{2} - 7(3 + 2k)\bigg] = 0$

$\rm :\longmapsto\:{(1 + 3k)}^{2} - 7(3 + 2k) = 0$

$\rm :\longmapsto\:1 + {9k}^{2} + 6k - 21 - 14k = 0$

$\rm :\longmapsto\:{9k}^{2} - 8k - 20 = 0$

$\rm :\longmapsto\:{9k}^{2} - 18k + 10k - 20 = 0$

$\rm :\longmapsto\:9k(k - 2) + 10(k - 2) = 0$

$\rm :\longmapsto\:(k - 2)(9k+ 10) = 0$

$\purple{\rm \implies\:\boxed{\tt{ k = 2 \: \: \: or \: \: \: k = - \dfrac{10}{9} }}}$

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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

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

Given quadratic equation is

$\rm :\longmapsto\: {x}^{2} - 2x(1 + 3k) + 7(3 + 2k) = 0$

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

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

$\red{\rm :\longmapsto\:b = - 2(1 + 3k)}$

$\red{\rm :\longmapsto\:c = 7(3 + 2k)}$

We know,

Quadratic equation ax² + bx + c = 0, have real and equal roots iff Discriminant, D = 0.

$\rm :\longmapsto\: {b}^{2} - 4ac = 0$

$\rm :\longmapsto\: {\bigg[ - 2(1 + 3k)\bigg]}^{2} - 4(1)[7(3 + 2k] = 0$

$\rm :\longmapsto\:4 {(1 + 3k)}^{2} - 28(3 + 2k) = 0$

$\rm :\longmapsto\:4 \bigg[{(1 + 3k)}^{2} - 7(3 + 2k)\bigg] = 0$

$\rm :\longmapsto\:{(1 + 3k)}^{2} - 7(3 + 2k) = 0$

$\rm :\longmapsto\:1 + {9k}^{2} + 6k - 21 - 14k = 0$

$\rm :\longmapsto\:{9k}^{2} - 8k - 20 = 0$

$\rm :\longmapsto\:{9k}^{2} - 18k + 10k - 20 = 0$

$\rm :\longmapsto\:9k(k - 2) + 10(k - 2) = 0$

$\rm :\longmapsto\:(k - 2)(9k+ 10) = 0$

$\purple{\rm \implies\:\boxed{\tt{ k = 2 \: \: \: or \: \: \: k = - \dfrac{10}{9} }}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

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