Which Is A Zero Of The Function F ( X ) = 4 X 2 + 35 X + 24 F(x) = 4x^2 + 35x + 24 F ( X ) = 4 X 2 + 35 X + 24 ?A. 3 4 \frac{3}{4} 4 3 ​ B. − 3 4 -\frac{3}{4} − 4 3 ​ C. 8 D. 4

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Introduction

In algebra, a zero of a function is a value of the variable that makes the function equal to zero. Finding the zeroes of a quadratic function is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the zeroes of the quadratic function f(x)=4x2+35x+24f(x) = 4x^2 + 35x + 24.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means it has the general form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants, and a≠0a \neq 0. The graph of a quadratic function is a parabola, which is a U-shaped curve. The zeroes of a quadratic function are the points where the parabola intersects the x-axis.

The Quadratic Formula

To find the zeroes of a quadratic function, we can use the quadratic formula, which is given by:

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

In this formula, aa, bb, and cc are the coefficients of the quadratic function, and xx is the variable. The quadratic formula gives us two solutions, which are the zeroes of the function.

Applying the Quadratic Formula to the Given Function

Now, let's apply the quadratic formula to the given function f(x)=4x2+35x+24f(x) = 4x^2 + 35x + 24. We have a=4a = 4, b=35b = 35, and c=24c = 24. Plugging these values into the quadratic formula, we get:

x=−35±352−4(4)(24)2(4)x = \frac{-35 \pm \sqrt{35^2 - 4(4)(24)}}{2(4)}

Simplifying the expression under the square root, we get:

x=−35±1225−3848x = \frac{-35 \pm \sqrt{1225 - 384}}{8}

x=−35±8418x = \frac{-35 \pm \sqrt{841}}{8}

x=−35±298x = \frac{-35 \pm 29}{8}

Finding the Zeroes

Now, we have two possible solutions:

x=−35+298x = \frac{-35 + 29}{8}

x=−68x = \frac{-6}{8}

x=−34x = -\frac{3}{4}

x=−35−298x = \frac{-35 - 29}{8}

x=−648x = \frac{-64}{8}

x=−8x = -8

Conclusion

In this article, we have found the zeroes of the quadratic function f(x)=4x2+35x+24f(x) = 4x^2 + 35x + 24 using the quadratic formula. We have obtained two solutions, which are x=−34x = -\frac{3}{4} and x=−8x = -8. These values make the function equal to zero, and they are the zeroes of the function.

Answer

The correct answer is B. −34-\frac{3}{4}.

Additional Tips and Tricks

  • When using the quadratic formula, make sure to simplify the expression under the square root.
  • When finding the zeroes of a quadratic function, make sure to check if the solutions are real or complex.
  • When graphing a quadratic function, make sure to plot the zeroes on the graph.

Real-World Applications

Finding the zeroes of a quadratic function has numerous real-world applications, including:

  • Physics: Finding the zeroes of a quadratic function can help us determine the position of an object in motion.
  • Engineering: Finding the zeroes of a quadratic function can help us design and optimize systems, such as bridges and buildings.
  • Economics: Finding the zeroes of a quadratic function can help us model and analyze economic systems, such as supply and demand curves.

Conclusion

In conclusion, finding the zeroes of a quadratic function is an essential skill in mathematics, and it has numerous real-world applications. By using the quadratic formula and simplifying the expression under the square root, we can find the zeroes of a quadratic function. We hope that this article has provided you with a better understanding of how to find the zeroes of a quadratic function and how to apply this skill in real-world scenarios.

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Introduction

In our previous article, we discussed how to find the zeroes of a quadratic function using the quadratic formula. In this article, we will provide a Q&A guide to help you better understand the concept of quadratic function zeroes and how to apply it in real-world scenarios.

Q&A

Q: What is a zero of a quadratic function?

A: A zero of a quadratic function is a value of the variable that makes the function equal to zero.

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

A: To find the zeroes 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 quadratic formula?

A: The quadratic formula is a mathematical formula that gives us the solutions to a quadratic equation. It is given by:

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

Q: What are the coefficients of a quadratic function?

A: The coefficients of a quadratic function are the numbers that are multiplied by the variable and the constant term. In the quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c, the coefficients are aa, bb, and cc.

Q: How do I simplify the expression under the square root in the quadratic formula?

A: To simplify the expression under the square root, you need to calculate the value of b2−4acb^2 - 4ac and then take the square root of that value.

Q: What are the real-world applications of finding the zeroes of a quadratic function?

A: Finding the zeroes of a quadratic function has numerous real-world applications, including:

  • Physics: Finding the zeroes of a quadratic function can help us determine the position of an object in motion.
  • Engineering: Finding the zeroes of a quadratic function can help us design and optimize systems, such as bridges and buildings.
  • Economics: Finding the zeroes of a quadratic function can help us model and analyze economic systems, such as supply and demand curves.

Q: Can I use the quadratic formula to find the zeroes of a quadratic function with complex solutions?

A: Yes, you can use the quadratic formula to find the zeroes of a quadratic function with complex solutions. However, you need to be careful when simplifying the expression under the square root, as it may involve complex numbers.

Q: How do I graph a quadratic function and plot its zeroes?

A: To graph a quadratic function and plot its zeroes, you need to use a graphing calculator or a computer software. You can also use a graphing app on your smartphone or tablet.

Conclusion

In conclusion, finding the zeroes of a quadratic function is an essential skill in mathematics, and it has numerous real-world applications. By using the quadratic formula and simplifying the expression under the square root, we can find the zeroes of a quadratic function. We hope that this Q&A guide has provided you with a better understanding of how to find the zeroes of a quadratic function and how to apply this skill in real-world scenarios.

Additional Tips and Tricks

  • When using the quadratic formula, make sure to simplify the expression under the square root.
  • When finding the zeroes of a quadratic function, make sure to check if the solutions are real or complex.
  • When graphing a quadratic function, make sure to plot the zeroes on the graph.

Real-World Examples

  • Physics: A ball is thrown upwards from the ground with an initial velocity of 20 m/s. The height of the ball above the ground at any time t is given by the quadratic function h(t)=−5t2+20th(t) = -5t^2 + 20t. Find the time at which the ball reaches its maximum height.
  • Engineering: A bridge is designed to span a river with a length of 100 meters. The weight of the bridge is given by the quadratic function w(x)=2x2+10x+20w(x) = 2x^2 + 10x + 20, where x is the distance from the center of the bridge. Find the distance from the center of the bridge at which the weight of the bridge is maximum.
  • Economics: The demand for a product is given by the quadratic function d(p)=−2p2+10p+20d(p) = -2p^2 + 10p + 20, where p is the price of the product. Find the price at which the demand for the product is maximum.

Conclusion

In conclusion, finding the zeroes of a quadratic function is an essential skill in mathematics, and it has numerous real-world applications. By using the quadratic formula and simplifying the expression under the square root, we can find the zeroes of a quadratic function. We hope that this Q&A guide has provided you with a better understanding of how to find the zeroes of a quadratic function and how to apply this skill in real-world scenarios.