Question

find the value of k for which the following equation has equal roots (k-12)x^2+2(k-12)+2=0​

Answer 1

EXPLANATION.

Equation has real and equal roots.

⇒ (k - 12)x² + 2(k - 12)x + 2 = 0.

As we know that,

For real and equal roots.

⇒ D = 0 Or b² - 4ac = 0.

Using this concept in the equation, we get.

⇒ [2(k - 12)²] - 4(k - 12)(2) = 0.

⇒ [4(k² + 144 - 24k)] - 8(k - 12) = 0.

⇒ [4k² + 576 - 96k] - 8k + 96 = 0.

⇒ 4k² + 576 - 96k - 8k + 96 = 0.

⇒ 4k² - 104k + 672 = 0.

⇒ 4(k² - 26k + 168) = 0.

⇒ k² - 26k + 168 = 0.

Factorizes the equation into middle term splits, we get.

⇒ k² - 14k - 12k + 168 = 0.

⇒ k(k - 14) - 12(k - 14) = 0.

⇒ (k - 12)(k - 14) = 0.

⇒ k = 12   and   k = 14.

                                                                                                                   

MORE INFORMATION.

Nature of roots of quadratic expression.

(1) Roots are real and unequal, if b² - 4ac > 0.

(2) Roots are rational and different, if b² - 4ac is a perfect square.

(3) Roots are real and equal, if b² - 4ac = 0.

(4) If D < 0 Roots are imaginary and unequal Or complex conjugate.

Answer 2

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EXPLANATION.

Equation has real and equal roots.

(k-12)x² + 2(k-12)x+ 2 = 0.

As we know that,

For real and equal roots.

→ D = 0 Or b² - 4ac = 0. Using this concept in the equation, we

get.

[2(k-12)³] 4(k-12)(2) = 0.

[4(k²+144-24k)]- 8(k - 12) = 0.

+ [4k² +576-96k] - 8k+96= 0. - 4k² +576-96k-8k+96= 0.

→ 4k² - 104k +672 = 0.

4(k²-26k+168) = 0. k²-26k + 168 = 0.

Factorizes the equation into middle term

splits, we get.

k²-14k-12k + 168 = 0,

k(k-14)-12(k-14) = 0.

(k-12)(k-14)=0.

k=12 and k = 14.

MORE INFORMATION.

Nature of roots of quadratic expression.

(1) Roots are real and unequal, if b²

0.

- 4ac >

(2) Roots are rational and different, if b² 4ac is a perfect square.

(3) Roots are real and equal, if b² - 4ac = 0.

(4) If D <0 Roots are imaginary and unequal or complex conjugate.