Which Are The Roots Of The Quadratic Function F ( B ) = B 2 − 75 F(b)=b^2-75 F ( B ) = B 2 − 75 ?Select Two Options.A. B = 5 3 B=5 \sqrt{3} B = 5 3 ​ B. B = − 5 3 B=-5 \sqrt{3} B = − 5 3 ​ C. B = 3 5 B=3 \sqrt{5} B = 3 5 ​ D. B = − 3 5 B=-3 \sqrt{5} B = − 3 5 ​ E. B = 25 3 B=25 \sqrt{3} B = 25 3 ​

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the roots of the quadratic function $f(b)=b^2-75$. We will delve into the world of quadratic equations, learn how to identify the roots, and apply this knowledge to solve the given function.

Understanding Quadratic Equations

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 $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The roots of a quadratic equation are the values of $x$ that satisfy the equation.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that the roots of the quadratic equation $ax^2+bx+c=0$ are given by:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Applying the Quadratic Formula to $f(b)=b^2-75$

To find the roots of the quadratic function $f(b)=b^2-75$, we can apply the quadratic formula. First, we need to rewrite the function in the standard form of a quadratic equation:

$$b^2-75=0$$

Now, we can identify the values of $a$, $b$, and $c$:

$$a=1, b=0, c=-75$$

Substituting the Values into the Quadratic Formula

Substituting the values of $a$, $b$, and $c$ into the quadratic formula, we get:

$$b=\frac{-0\pm\sqrt{0^2-4(1)(-75)}}{2(1)}$$

Simplifying the Expression

Simplifying the expression, we get:

$$b=\frac{\pm\sqrt{300}}{2}$$

Simplifying the Square Root

Simplifying the square root, we get:

$$b=\frac{\pm10\sqrt{3}}{2}$$

Simplifying the Expression Further

Simplifying the expression further, we get:

$$b=\pm5\sqrt{3}$$

Conclusion

In conclusion, the roots of the quadratic function $f(b)=b^2-75$ are $b=5\sqrt{3}$ and $b=-5\sqrt{3}$. These values satisfy the equation and are the solutions to the quadratic function.

Answer Options

Based on our calculations, the correct answer options are:

• A. $b=5\sqrt{3}$
• B. $b=-5\sqrt{3}$

The other options are incorrect, as they do not satisfy the equation.

Final Thoughts

Solving quadratic equations is an essential skill for students to master. By applying the quadratic formula and simplifying the expression, we can find the roots of the quadratic function $f(b)=b^2-75$. This knowledge can be applied to a wide range of mathematical problems and is a fundamental concept in mathematics.

Additional Resources

For further learning, we recommend exploring the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Equation Solver

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline
  

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Introduction

In our previous article, we explored the roots of the quadratic function $f(b)=b^2-75$. We applied the quadratic formula and simplified the expression to find the solutions. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights into solving quadratic functions.

Q&A

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that the roots of the quadratic equation $ax^2+bx+c=0$ are given by:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Q: How do I apply the quadratic formula to a quadratic function?

A: To apply the quadratic formula to a quadratic function, you need to rewrite the function in the standard form of a quadratic equation. Then, you can identify the values of $a$, $b$, and $c$ and substitute them into the quadratic formula.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. The quadratic formula is a general method that can be applied to any quadratic equation, while factoring is a specific method that can be used to solve quadratic equations that can be factored.

Q: Can I use the quadratic formula to solve quadratic functions with complex roots?

A: Yes, you can use the quadratic formula to solve quadratic functions with complex roots. The quadratic formula will give you the complex roots of the equation.

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

A: To simplify the expression in the quadratic formula, you need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Exponentiate the expressions (e.g., square roots).
3. Multiply and divide the expressions from left to right.
4. Add and subtract the expressions from left to right.

Q: Can I use the quadratic formula to solve quadratic functions with rational roots?

A: Yes, you can use the quadratic formula to solve quadratic functions with rational roots. The quadratic formula will give you the rational roots of the equation.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant ($b^2-4ac$). If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By applying the quadratic formula and simplifying the expression, we can find the roots of the quadratic function $f(b)=b^2-75$. We hope this Q&A article has provided additional insights into solving quadratic functions and has helped you to better understand the quadratic formula.

Additional Resources

For further learning, we recommend exploring the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Equation Solver

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline