How Many Real Zeros Does This Quadratic Function Have?$y = X^2 + 6x + 8$Enter Your Answer In The Box.

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Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants, and aa cannot be equal to zero. In this article, we will focus on the quadratic function y=x2+6x+8y = x^2 + 6x + 8 and determine the number of real zeros it has.

What are Real Zeros?

Real zeros of a quadratic function are the values of xx that make the function equal to zero. In other words, they are the points where the graph of the function intersects the x-axis. To find the real zeros of a quadratic function, we need to solve the equation ax2+bx+c=0ax^2 + bx + c = 0.

The Quadratic Formula

The quadratic formula is a powerful tool for finding the real zeros of a quadratic function. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula works by first calculating the discriminant, which is the expression under the square root. If the discriminant is positive, then the quadratic function has two distinct real zeros. If the discriminant is zero, then the quadratic function has one real zero. If the discriminant is negative, then the quadratic function has no real zeros.

Applying the Quadratic Formula to the Given Function

Now, let's apply the quadratic formula to the given function y=x2+6x+8y = x^2 + 6x + 8. We have a=1a = 1, b=6b = 6, and c=8c = 8. Plugging these values into the quadratic formula, we get:

x=−6±62−4(1)(8)2(1)x = \frac{-6 \pm \sqrt{6^2 - 4(1)(8)}}{2(1)}

Simplifying the expression under the square root, we get:

x=−6±36−322x = \frac{-6 \pm \sqrt{36 - 32}}{2}

x=−6±42x = \frac{-6 \pm \sqrt{4}}{2}

x=−6±22x = \frac{-6 \pm 2}{2}

Solving for the Real Zeros

Now, we have two possible values for xx:

x=−6+22x = \frac{-6 + 2}{2}

x=−42x = \frac{-4}{2}

x=−2x = -2

x=−6−22x = \frac{-6 - 2}{2}

x=−82x = \frac{-8}{2}

x=−4x = -4

Conclusion

In conclusion, the quadratic function y=x2+6x+8y = x^2 + 6x + 8 has two real zeros, which are x=−2x = -2 and x=−4x = -4. These values of xx make the function equal to zero, and they are the points where the graph of the function intersects the x-axis.

Real-World Applications

Quadratic functions have many real-world applications, including:

  • Projectile Motion: The trajectory of a projectile under the influence of gravity can be modeled using a quadratic function.
  • Optimization: Quadratic functions can be used to optimize functions, such as finding the maximum or minimum value of a function.
  • Statistics: Quadratic functions can be used to model the relationship between two variables, such as the relationship between the price of a product and its demand.

Final Thoughts

In this article, we have seen how to find the real zeros of a quadratic function using the quadratic formula. We have also seen some real-world applications of quadratic functions. Quadratic functions are an important part of mathematics, and they have many practical applications in fields such as physics, engineering, and economics.

Glossary of Terms

  • Quadratic Function: A polynomial function of degree two, which means the highest power of the variable is two.
  • Real Zeros: The values of xx that make the function equal to zero.
  • Discriminant: The expression under the square root in the quadratic formula.
  • Quadratic Formula: A powerful tool for finding the real zeros of a quadratic function.

References

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak
  • "Quadratic Equations" by Math Open Reference

Note: The references provided are for general information purposes only and are not specific to the topic of this article.

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Introduction

In our previous article, we discussed the quadratic function y=x2+6x+8y = x^2 + 6x + 8 and determined the number of real zeros it has. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants, and aa cannot be equal to zero.

Q: What are real zeros?

A: Real zeros of a quadratic function are the values of xx that make the function equal to zero. In other words, they are the points where the graph of the function intersects the x-axis.

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

A: To find the real zeros of a quadratic function, you can use the quadratic formula, which is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the discriminant?

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

Q: Can I use the quadratic formula to find the real zeros of a quadratic function with a negative discriminant?

A: No, you cannot use the quadratic formula to find the real zeros of a quadratic function with a negative discriminant. In this case, the quadratic function has no real zeros.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including:

  • Projectile Motion: The trajectory of a projectile under the influence of gravity can be modeled using a quadratic function.
  • Optimization: Quadratic functions can be used to optimize functions, such as finding the maximum or minimum value of a function.
  • Statistics: Quadratic functions can be used to model the relationship between two variables, such as the relationship between the price of a product and its demand.

Q: Can I use a calculator to find the real zeros of a quadratic function?

A: Yes, you can use a calculator to find the real zeros of a quadratic function. Most calculators have a built-in quadratic formula function that you can use to find the real zeros of a quadratic function.

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

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. In other words, a quadratic function has a squared term, while a linear function does not.

Q: Can I use a quadratic function to model a non-linear relationship between two variables?

A: Yes, you can use a quadratic function to model a non-linear relationship between two variables. However, you need to be careful when using a quadratic function to model a non-linear relationship, as it may not accurately capture the relationship between the two variables.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

  • Not checking the discriminant: Make sure to check the discriminant before using the quadratic formula to find the real zeros of a quadratic function.
  • Not using the correct formula: Make sure to use the correct formula for the quadratic function, including the correct values for aa, bb, and cc.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions when solving a quadratic equation.

Conclusion

In conclusion, quadratic functions are an important part of mathematics, and they have many practical applications in fields such as physics, engineering, and economics. By understanding the quadratic function and its properties, you can use it to model a wide range of real-world phenomena.

Glossary of Terms

  • Quadratic Function: A polynomial function of degree two, which means the highest power of the variable is two.
  • Real Zeros: The values of xx that make the function equal to zero.
  • Discriminant: The expression under the square root in the quadratic formula.
  • Quadratic Formula: A powerful tool for finding the real zeros of a quadratic function.

References

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak
  • "Quadratic Equations" by Math Open Reference

Note: The references provided are for general information purposes only and are not specific to the topic of this article.