Question

Find the value of k for which the quadratic
equation (k - 12) xº + 2 (k - 12) x + 2 - 0 has
real and equal roots.

​

Answer 1

Required Answer:-

Question:

Find the value of k for which the quadratic equation (k - 12)x² + 2(k - 12)x + 2 = 0 has real and equal roots.

Solution:

We have,

➡ (k - 12)x² + 2(k - 12)x + 2 = 0

Here,

1. a (coefficient of x²) = k - 12
2. b (coefficient of x) = 2(k - 12)
3. c (coefficient of x⁰) = 2

★ The discriminant of a quadratic equation tells the nature of roots. If the discriminant is equal to zero, the roots of the Quadratic equation are real and equal.

Discriminant is calculated by using the formula,

➡ D = b² - 4ac.

So, according to the given condition,

➡ b² - 4ac = 0

➡ [2(k - 12)]² - 4 × (k - 12) × 2 = 0

➡ (2k - 24)² - 8(k - 12) = 0

➡ 4k² + 24² - 2 × 2k × 24 - 8(k - 12) = 0

➡ 4k² + 576 - 96k - 8k + 96 = 0

➡ 4k² - 104k + 672 = 0

➡ 4(k² - 26k + 168) = 0

➡ k² - 26k + 168 = 0

➡ k² - 12k - 14k + 168 = 0

➡ k(k - 12) - 14(k - 12) = 0

➡ (k - 14)(k - 12) = 0

By zero product rule,

➡ Either k - 14 = 0 or k - 12 = 0

➡ k = 12, 14

But we found that, when k = 12, there exists one root for the equation and not two. So, we omit k = 12.

➡ k = 14

★ Hence, the values of k is 14 for which the given equation has real and equal roots.

Verification:

Let us verify our result.

When k = 14,

➡ (k - 12)x² - 2(k - 12)x + 2 = 0

➡ 2x² - 2 × 2x + 2 = 0

➡ 2(x² - 2x + 1) = 0

➡ x² - 2x + 1 = 0

On solving, we get x = 1,1

So, the value of k is 14 for which the given equation has two real and equal roots. (Verified)

Answer:

• The value of k is 14.

Answer 2

A quadratic equation is given which has real and equal roots.

• (k - 12) x² + 2 (k - 12) + 2 = 0

As we know that, if a quadratic equation has real and equal roots, then discriminant = 0.

Discriminant = b² - 4ac

Where

• a = (k - 12), which is coefficient of x²

• b = 2 (k - 12), which is coefficient of x

• c = 2, which is constant term

★ Put the values in the formula

➟ b² - 4ac = 0

➟ [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 + 96 - 96k - 8k = 0

➟ 4k² + 672 - 104k = 0

➟ k² + 168 - 26k = 0

It's in the form of quadratic equation

➟ k² - 26k + 168 = 0

➟ k² - 14k - 12k + 168 = 0

➟ k (k - 14) - 12 (k - 14) = 0

➟ (k - 14) (k - 12)

________________

➟ k - 14 = 0

➟ k = 14

________________

➟ k - 12 = 0

➟ k = 12

________________

.°. The value of k is 14 and 12.