From the uploaded image, the question asks to find the correct statements about the complete quadratic equation $ax^2 + bx + c = 0$ when $D = 0$, where $D$ is the discriminant given by $D = b^2 - 4ac$.

Analyzing the Problem

When $D = 0$, the quadratic equation has exactly one repeated (real) root. This happens because the discriminant determines the nature of the roots:

• If $D > 0$: Two distinct real roots.
• If $D = 0$: Two equal real roots (repeated root).
• If $D < 0$: No real solutions (complex roots).

The roots in case of $D = 0$ are calculated as: $x = \frac{-b}{2a}$

Correct Statements

Based on this, the correct options would be:

• b: "The equation has only one real solution." (This is correct since the repeated root counts as one unique solution.)
• c: "The equation has two equal real solutions." (This is correct since the equation has a double root.)
• d: $x = \frac{-b}{2a}$. (This matches the formula for the root when $D = 0$.)

Other options:

• a: Incorrect, because there is a real solution when $D = 0$.
• e: Incorrect, because the equation does not have two distinct real solutions when $D = 0$.
• f, g, h, i: These provide incorrect formulas for the roots and do not apply.

Would you like further clarification or a breakdown of any specific option?

Related Questions

1. What is the relationship between the discriminant and the nature of roots in a quadratic equation?
2. How is the formula for the roots derived in a quadratic equation?
3. Can you explain why $D = 0$ leads to a repeated root geometrically?
4. What happens to the graph of $ax^2 + bx + c = 0$ when $D = 0$?
5. How does the value of $a$ influence the roots of a quadratic equation?

Tip:

Always start by calculating the discriminant ($D$) before determining the nature of the roots for any quadratic equation.