(E) 7. If X² + Bx+b=0 Has Two Real And Distinct Roots, Then The Value Of B Can Be (A) 0 (C) 3 (B) 4 (D) -3
Introduction
In algebra, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In this article, we will focus on the quadratic equation x² + bx + b = 0 and explore the conditions under which it has two real and distinct roots.
The Nature of Roots in a Quadratic Equation
The roots of a quadratic equation are the values of the variable that satisfy the equation. In the case of a quadratic equation with real coefficients, the roots can be real or complex. However, if the equation has two real and distinct roots, it means that the roots are two different real numbers.
The Discriminant: A Key to Understanding the Nature of Roots
The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is given by the formula D = b² - 4ac. The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation. If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.
Applying the Discriminant to the Equation x² + bx + b = 0
To determine the value of b for which the equation x² + bx + b = 0 has two real and distinct roots, we need to calculate the discriminant. Since the equation is x² + bx + b = 0, we have a = 1, b = b, and c = b. Plugging these values into the formula for the discriminant, we get D = b² - 4(1)(b) = b² - 4b.
Solving the Inequality b² - 4b > 0
For the equation x² + bx + b = 0 to have two real and distinct roots, the discriminant must be positive. Therefore, we need to solve the inequality b² - 4b > 0. To solve this inequality, we can factor the left-hand side as b(b - 4) > 0. This inequality is true when either b > 0 and b - 4 > 0, or b < 0 and b - 4 < 0.
Solving the Inequality b > 0 and b - 4 > 0
To solve the inequality b > 0 and b - 4 > 0, we can first solve the inequality b - 4 > 0. This inequality is true when b > 4. Since b > 0, we can conclude that b > 4.
Solving the Inequality b < 0 and b - 4 < 0
To solve the inequality b < 0 and b - 4 < 0, we can first solve the inequality b - 4 < 0. This inequality is true when b < 4. Since b < 0, we can conclude that b < 0 and b < 4.
Combining the Results
Combining the results of the previous two inequalities, we can conclude that the value of b for which the equation x² + bx + b = 0 has two real and distinct roots is b > 4 or b < 0 and b < 4.
Conclusion
In conclusion, the value of b for which the equation x² + bx + b = 0 has two real and distinct roots is b > 4 or b < 0 and b < 4. This means that the correct answer is not among the options provided in the question. However, if we need to choose an answer from the options provided, we can choose the option that is closest to the correct answer.
Final Answer
The final answer is:
Q: What is the discriminant of a quadratic equation?
A: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is given by the formula D = b² - 4ac. The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation.
Q: How do I determine the nature of the roots of a quadratic equation?
A: To determine the nature of the roots of a quadratic equation, you need to calculate the discriminant. If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.
Q: What is the significance of the discriminant in a quadratic equation?
A: The discriminant is significant in a quadratic equation because it determines the nature of the roots. If the discriminant is positive, the equation has two real and distinct roots, which means that the roots are two different real numbers. If the discriminant is zero, the equation has one real root, which means that the root is a single real number. If the discriminant is negative, the equation has two complex roots, which means that the roots are two complex numbers.
Q: How do I calculate the discriminant of a quadratic equation?
A: To calculate the discriminant of a quadratic equation, you need to use the formula D = b² - 4ac. This formula requires you to know the coefficients of the equation, which are a, b, and c.
Q: What is the relationship between the discriminant and the roots of a quadratic equation?
A: The discriminant is related to the roots of a quadratic equation in the following way: if the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.
Q: Can the discriminant be zero?
A: Yes, the discriminant can be zero. This occurs when the equation has one real root. In this case, the discriminant is equal to zero, and the equation has a single real root.
Q: Can the discriminant be negative?
A: Yes, the discriminant can be negative. This occurs when the equation has two complex roots. In this case, the discriminant is negative, and the equation has two complex roots.
Q: What is the significance of the value of b in a quadratic equation?
A: The value of b in a quadratic equation is significant because it determines the nature of the roots. If the value of b is greater than 4, the equation has two real and distinct roots. If the value of b is less than 0 and less than 4, the equation has two real and distinct roots.
Q: Can the value of b be 0?
A: No, the value of b cannot be 0. This is because if the value of b is 0, the equation has no real roots.
Q: Can the value of b be 3?
A: No, the value of b cannot be 3. This is because if the value of b is 3, the equation has no real roots.
Q: Can the value of b be -3?
A: No, the value of b cannot be -3. This is because if the value of b is -3, the equation has no real roots.
Q: Can the value of b be 4?
A: Yes, the value of b can be 4. This is because if the value of b is 4, the equation has two real and distinct roots.
Conclusion
In conclusion, the discriminant is a crucial value in a quadratic equation that determines the nature of the roots. The value of b also plays a significant role in determining the nature of the roots. By understanding the relationship between the discriminant and the roots, you can solve quadratic equations and determine the nature of the roots.