How Many Real Zeros Does The Following Quadratic Function Have? F ( X ) = X 2 − 3 X + 27 F(x) = X^2 - 3x + 27 F ( X ) = X 2 − 3 X + 27 A. One Real Zero B. No Real Zeros C. Two Real Zeros

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How Many Real Zeros Does the Quadratic Function Have?

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form of f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards. In this article, we will focus on finding the number of real zeros of a quadratic function.

The Quadratic Formula

To find the zeros of a quadratic function, we can use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. This formula gives us two possible values for xx, which are the zeros of the function.

The Given Quadratic Function

The given quadratic function is f(x)=x2−3x+27f(x) = x^2 - 3x + 27. To find the number of real zeros of this function, we can use the quadratic formula.

Finding the Discriminant

The discriminant of a quadratic function is the expression under the square root in the quadratic formula: b2−4acb^2 - 4ac. If the discriminant is positive, then the function has two real zeros. If the discriminant is zero, then the function has one real zero. If the discriminant is negative, then the function has no real zeros.

Calculating the Discriminant

To calculate the discriminant of the given quadratic function, we need to substitute the values of aa, bb, and cc into the expression: b2−4acb^2 - 4ac. In this case, a=1a = 1, b=−3b = -3, and c=27c = 27. Therefore, the discriminant is (−3)2−4(1)(27)=9−108=−99(-3)^2 - 4(1)(27) = 9 - 108 = -99.

Interpreting the Discriminant

Since the discriminant is negative, we can conclude that the given quadratic function has no real zeros.

Conclusion

In conclusion, the quadratic function f(x)=x2−3x+27f(x) = x^2 - 3x + 27 has no real zeros. This is because the discriminant of the function is negative, which means that the function does not intersect the x-axis at any point.

Real-World Applications

Understanding the number of real zeros of a quadratic function has many real-world applications. For example, in physics, the motion of an object can be modeled using a quadratic function. The zeros of the function represent the points in time when the object is at rest. In engineering, the design of a bridge can be modeled using a quadratic function. The zeros of the function represent the points where the bridge is at its maximum height.

Tips and Tricks

When working with quadratic functions, it is essential to remember the following tips and tricks:

  • The discriminant of a quadratic function can be used to determine the number of real zeros of the function.
  • If the discriminant is positive, then the function has two real zeros.
  • If the discriminant is zero, then the function has one real zero.
  • If the discriminant is negative, then the function has no real zeros.

Common Mistakes

When working with quadratic functions, it is common to make the following mistakes:

  • Failing to calculate the discriminant correctly.
  • Misinterpreting the discriminant.
  • Not using the quadratic formula to find the zeros of the function.

Conclusion

In conclusion, the quadratic function f(x)=x2−3x+27f(x) = x^2 - 3x + 27 has no real zeros. This is because the discriminant of the function is negative, which means that the function does not intersect the x-axis at any point. Understanding the number of real zeros of a quadratic function has many real-world applications, and it is essential to remember the tips and tricks for working with quadratic functions.

Final Thoughts

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is essential for solving many real-world problems. In this article, we have discussed how to find the number of real zeros of a quadratic function using the discriminant. We have also provided tips and tricks for working with quadratic functions and common mistakes to avoid. By following these guidelines, you can become proficient in working with quadratic functions and apply your knowledge to real-world problems.

References

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form of f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards. In this article, we will focus on answering frequently asked questions about quadratic function zeros.

Q: What is a quadratic function zero?

A: A quadratic function zero is a value of xx that makes the function equal to zero. In other words, it is a point on the x-axis where the function intersects the x-axis.

Q: How do I find the zeros of a quadratic function?

A: To find the zeros of a quadratic function, you can use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. This formula gives you two possible values for xx, which are the zeros of the function.

Q: What is the discriminant of a quadratic function?

A: The discriminant of a quadratic function is the expression under the square root in the quadratic formula: b2−4acb^2 - 4ac. If the discriminant is positive, then the function has two real zeros. If the discriminant is zero, then the function has one real zero. If the discriminant is negative, then the function has no real zeros.

Q: How do I calculate the discriminant of a quadratic function?

A: To calculate the discriminant of a quadratic function, you need to substitute the values of aa, bb, and cc into the expression: b2−4acb^2 - 4ac. For example, if the quadratic function is f(x)=x2−3x+27f(x) = x^2 - 3x + 27, then the discriminant is (−3)2−4(1)(27)=9−108=−99(-3)^2 - 4(1)(27) = 9 - 108 = -99.

Q: What does the discriminant tell me about the zeros of a quadratic function?

A: The discriminant tells you the number of real zeros of a quadratic function. If the discriminant is positive, then the function has two real zeros. If the discriminant is zero, then the function has one real zero. If the discriminant is negative, then the function has no real zeros.

Q: Can a quadratic function have complex zeros?

A: Yes, a quadratic function can have complex zeros. If the discriminant is negative, then the function has complex zeros. Complex zeros are values of xx that are not real numbers.

Q: How do I find the complex zeros of a quadratic function?

A: To find the complex zeros of a quadratic function, you can use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. If the discriminant is negative, then the expression under the square root will be a negative number, which will give you complex zeros.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree 2, while a linear function is a polynomial function of degree 1. A quadratic function has a parabolic graph, while a linear function has a straight line graph.

Q: Can a quadratic function have a zero at x=0x = 0?

A: Yes, a quadratic function can have a zero at x=0x = 0. This is called a zero at the origin.

Q: How do I determine if a quadratic function has a zero at x=0x = 0?

A: To determine if a quadratic function has a zero at x=0x = 0, you can substitute x=0x = 0 into the function and see if it equals zero. If it does, then the function has a zero at x=0x = 0.

Q: Can a quadratic function have multiple zeros?

A: Yes, a quadratic function can have multiple zeros. This is called a repeated zero.

Q: How do I determine if a quadratic function has multiple zeros?

A: To determine if a quadratic function has multiple zeros, you can use the discriminant. If the discriminant is zero, then the function has a repeated zero.

Conclusion

In conclusion, quadratic function zeros are an essential concept in mathematics. Understanding how to find the zeros of a quadratic function, how to calculate the discriminant, and how to determine the number of real zeros is crucial for solving many real-world problems. By following the tips and tricks outlined in this article, you can become proficient in working with quadratic functions and apply your knowledge to real-world problems.

Final Thoughts

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is essential for solving many real-world problems. In this article, we have discussed frequently asked questions about quadratic function zeros, including how to find the zeros of a quadratic function, how to calculate the discriminant, and how to determine the number of real zeros. By following the guidelines outlined in this article, you can become proficient in working with quadratic functions and apply your knowledge to real-world problems.

References