Question

It is given that the roots of the equation x2 – 2x – log2 K = 0 are real. For this, the minimum value of K is
1. 1
2. 1/2
3. 1/4
4. 1/16

Answer 1

Correct Answer - Option 2 : 1/2

Concept:

Let us consider the standard form of a quadratic equation, ax2 + bx + c =0

Discriminant = D = b2 – 4ac

• If the Discriminant > 0 then the roots are real and distinct.
• If the Discriminant = 0 then the roots are real and equal.
• If the Discriminant < 0 then the roots are Imaginary.

Logarithmic identities:

log_b(x) = c \(\rm \Leftrightarrow \) b^c = x

Calculation:

Given: x2 – 2x – log2 K = 0

General form of equation: ax² + bx + c = 0

On comparing a = 1, b = - 2 c = -  log2 K 

Given that the roots are real

So, Determinant ≥ 0

⇒ b2 – 4ac ≥ 0

⇒ (-2)² - 4 × 1 × (- log2 K) ≥  0

⇒ 4 + 4 log2 K ≥  0

⇒ 4 log2 K ≥  -4

⇒ log2 K ≥  -1

⇒ K  ≥  (2)^-1

∴ K ≥  1/2

The minimum value of K is 1/2.

 

Application of Quadratic Equation:

There are many real-world situations that deal with quadratics: Throwing a ball, shooting a cannon, diving from a platform and hitting a golf ball are all examples of situations that can be modeled by quadratic functions. 

Why quadratic equation is known as "quadratic"??

• The name quadratic comes from "quad" means square, because the variable gets squared (like x²).
• It is also called as "equation of degree 2" [because of 2 on the x]

 

Nature of Roots of a quadratic equation:

If the value of discriminant = 0 i.e. b2 – 4ac = 0	The quadratic equation will have equal roots i.e. α = β = - b/2a
If the value of discriminant < 0 i.e. b2 – 4ac < 0	The quadratic equation will have imaginary roots i.e α = (p + iq) and β = (p – iq). Where ‘iq’ is the imaginary part of a complex number
If the value of discriminant > 0  i.e. b2 – 4ac > 0	The quadratic equation will have real roots
If the value of discriminant > 0 and D is a perfect square	The quadratic equation will have rational roots
If the value of discriminant > 0 and D is not a perfect square	The quadratic equation will have irrational roots i.e. α = (p + √q) and β = (p – √q)
If the value of discriminant > 0, D is a perfect square, a = 1 and b and c are integers	The quadratic equation will have integral roots.