Finding the Zeroes of the Quadratic Equation

Oct 13, 2025 by miodeya.com 116 views

Introduction

In this problem, we are given a quadratic equation p(x) = x² - 3x + 4 and asked to find the value of (a+1)(B+1) where a and B are the zeroes of the equation. To solve this problem, we will first find the values of a and B using the quadratic formula, and then substitute these values into the expression (a+1)(B+1).

Quadratic Formula

The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation of the form ax² + bx + c = 0. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

In this case, the quadratic equation is p(x) = x² - 3x + 4, which can be written in the form ax² + bx + c = 0 as:

x² - 3x + 4 = 0

Comparing this with the general form of a quadratic equation, we have:

a = 1, b = -3, and c = 4

Finding the Zeroes of the Quadratic Equation

Now that we have the values of a, b, and c, we can use the quadratic formula to find the zeroes of the equation. Plugging in the values of a, b, and c into the quadratic formula, we get:

x = (3 ± √((-3)² - 4(1)(4))) / 2(1) x = (3 ± √(9 - 16)) / 2 x = (3 ± √(-7)) / 2

Since the square root of a negative number is an imaginary number, the zeroes of the equation are complex numbers. Let's call the two zeroes a and B.

Simplifying the Zeroes

We can simplify the zeroes by using the fact that √(-7) = √7i, where i is the imaginary unit. So, we can write:

a = (3 + √7i) / 2 B = (3 - √7i) / 2

Finding the Value of (a+1)(B+1)

Now that we have the values of a and B, we can substitute them into the expression (a+1)(B+1). We get:

(a+1)(B+1) = ((3 + √7i) / 2 + 1)((3 - √7i) / 2 + 1)

Simplifying the Expression

To simplify the expression, we can multiply out the brackets and combine like terms. We get:

(a+1)(B+1) = ((3 + √7i) / 2 + 1)((3 - √7i) / 2 + 1) = ((3 + √7i + 2) / 2)((3 - √7i + 2) / 2) = ((5 + √7i) / 2)((5 - √7i) / 2) = ((5 + √7i)(5 - √7i)) / 4 = (25 - 7i²) / 4 = (25 + 7) / 4 = 32 / 4 = 8

Conclusion

In this problem, we were given a quadratic equation p(x) = x² - 3x + 4 and asked to find the value of (a+1)(B+1) where a and B are the zeroes of the equation. We found the values of a and B using the quadratic formula, and then substituted these values into the expression (a+1)(B+1). After simplifying the expression, we found that the value of (a+1)(B+1) is 8.

Final Answer

The final answer is 8.

Introduction

In our previous article, we solved the problem of finding the value of (a+1)(B+1) where a and B are the zeroes of the quadratic equation p(x) = x² - 3x + 4. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the quadratic formula and how is it used to find the zeroes of a quadratic equation?

A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation of the form ax² + bx + c = 0. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

To find the zeroes of a quadratic equation, we plug in the values of a, b, and c into the quadratic formula and simplify the expression.

Q: What are the values of a, b, and c in the quadratic equation p(x) = x² - 3x + 4?

A: In the quadratic equation p(x) = x² - 3x + 4, we have:

a = 1, b = -3, and c = 4

Q: How do we simplify the zeroes of the quadratic equation?

A: To simplify the zeroes, we use the fact that √(-7) = √7i, where i is the imaginary unit. So, we can write:

a = (3 + √7i) / 2 B = (3 - √7i) / 2

Q: How do we find the value of (a+1)(B+1)?

A: To find the value of (a+1)(B+1), we substitute the values of a and B into the expression and simplify the result.

Q: What is the final answer to the problem?

A: The final answer to the problem is 8.

Q: Can you provide more information on the quadratic formula and how it is used to find the zeroes of a quadratic equation?

A: The quadratic formula is a powerful tool for finding the zeroes of a quadratic equation. It is used by plugging in the values of a, b, and c into the formula and simplifying the expression. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

This formula can be used to find the zeroes of any quadratic equation in the form ax² + bx + c = 0.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

Not plugging in the correct values of a, b, and c into the formula
Not simplifying the expression correctly
Not using the correct formula for the quadratic equation

Conclusion

In this Q&A article, we provided additional information and clarification on the problem of finding the value of (a+1)(B+1) where a and B are the zeroes of the quadratic equation p(x) = x² - 3x + 4. We hope that this article has been helpful in providing a better understanding of the topic.

Final Answer

The final answer to the problem is 8.