Question

Which of the following quadratic equations has real and equal roots?

• A. $x^2 + x = 0$
• B. $(x + 1)^2 = 2x + 1$
• C. $x^2 - 4 = 0$
• D. $x^2 + x + 1 = 0$

Hint:

Use the discriminant formula to determine root nature quickly.

Answer 1

The Correct Option is B

To Find:
Which of the following quadratic equations has real and equal roots?

Concept:
A quadratic equation has real and equal roots if its discriminant \(D = b^2 - 4ac = 0\)

Let's check each option: Option B:
\[ (x + 1)^2 = 2x + 1 \Rightarrow x^2 + 2x + 1 = 2x + 1 \Rightarrow x^2 + 2x + 1 - 2x - 1 = 0 \Rightarrow x^2 = 0 \Rightarrow \text{Only one root: } x = 0 \Rightarrow \text{Real and equal} \]
✅ This equation has real and equal roots.

Option A:
\[ x^2 + x = 0 \Rightarrow x(x + 1) = 0 \Rightarrow x = 0, x = -1 \Rightarrow \text{Two distinct real roots} \]
❌ Not equal

Option C:
\[ x^2 - 4 = 0 \Rightarrow x = \pm 2 \Rightarrow \text{Two distinct real roots} \]
❌ Not equal

Option D:
\[ x^2 + x + 1 = 0 \Rightarrow D = 1^2 - 4(1)(1) = 1 - 4 = -3 \Rightarrow \text{Imaginary roots} \]
❌ Not real

Final Answer:
✅ The correct option is B. \((x + 1)^2 = 2x + 1\)