Select The Correct Answer.Consider The Quadratic Equation Below.$\[ 4x^2 - 5 = 3x + 4 \\]Determine The Correct Set-up For Solving The Equation Using The Quadratic Formula.A. $\[ X = \frac{-(3) \pm \sqrt{(3)^2 - 4(-4)(-9)}}{2(-4)}

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving quadratic equations using the quadratic formula. We will explore the correct set-up for solving the equation and provide a step-by-step guide on how to use the quadratic formula.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) 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 Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Step 1: Identify the Coefficients

To use the quadratic formula, we need to identify the coefficients a, b, and c in the quadratic equation. In the given equation:

4x^2 - 5 = 3x + 4

we can rewrite it in the standard form as:

4x^2 - 3x - 9 = 0

Now, we can identify the coefficients:

a = 4 b = -3 c = -9

Step 2: Plug in the Coefficients

Now that we have identified the coefficients, we can plug them into the quadratic formula:

x = (-( -3 ) ± √( (-3)^2 - 4(4)(-9) )) / 2(4)

Step 3: Simplify the Expression

To simplify the expression, we can start by evaluating the expressions inside the square root:

(-3)^2 = 9 4(4)(-9) = -144

Now, we can substitute these values back into the expression:

x = (3 ± √(9 - (-144))) / 8

x = (3 ± √(153)) / 8

Conclusion

In this article, we have learned how to set up a quadratic equation for solving using the quadratic formula. We have identified the coefficients a, b, and c, plugged them into the quadratic formula, and simplified the expression. The correct set-up for solving the equation is:

x = (-(3) ± √((3)^2 - 4(-4)(-9))) / 2(-4)

This is the correct answer.

Discussion

  • What are some common mistakes students make when solving quadratic equations?
  • How can we use the quadratic formula to solve quadratic equations with complex roots?
  • Can we use the quadratic formula to solve quadratic equations with rational roots?

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Quadratic Formula" by Khan Academy
  • [3] "Solving Quadratic Equations" by Purplemath

Additional Resources

  • [1] "Quadratic Equations" by MIT OpenCourseWare
  • [2] "Quadratic Formula" by Wolfram Alpha
  • [3] "Solving Quadratic Equations" by IXL
    Quadratic Equations Q&A: Frequently Asked Questions =====================================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations and provide detailed answers to help you better understand this topic.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) 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.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I identify the coefficients in a quadratic equation?

A: To identify the coefficients, you need to rewrite the quadratic equation in the standard form:

ax^2 + bx + c = 0

Then, you can identify the coefficients a, b, and c.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. The general form of a linear equation is:

ax + b = 0

where a and b are constants, and x is the variable.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex roots. However, you need to be careful when simplifying the expression, as complex roots may involve imaginary numbers.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with rational roots. However, you need to be careful when simplifying the expression, as rational roots may involve fractions.

Q: What are some common mistakes students make when solving quadratic equations?

A: Some common mistakes students make when solving quadratic equations include:

  • Not rewriting the quadratic equation in the standard form
  • Not identifying the coefficients correctly
  • Not simplifying the expression correctly
  • Not using the correct formula (quadratic formula or factoring)

Q: How can I use the quadratic formula to solve quadratic equations with multiple solutions?

A: To solve quadratic equations with multiple solutions, you need to use the quadratic formula and simplify the expression. If the expression under the square root is a perfect square, you may have multiple solutions.

Q: Can I use the quadratic formula to solve quadratic equations with no solutions?

A: Yes, you can use the quadratic formula to solve quadratic equations with no solutions. If the expression under the square root is negative, you may have no solutions.

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided detailed answers to help you better understand this topic. We hope this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Discussion

  • What are some other common mistakes students make when solving quadratic equations?
  • How can we use the quadratic formula to solve quadratic equations with multiple solutions?
  • Can we use the quadratic formula to solve quadratic equations with no solutions?

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Quadratic Formula" by Khan Academy
  • [3] "Solving Quadratic Equations" by Purplemath

Additional Resources

  • [1] "Quadratic Equations" by MIT OpenCourseWare
  • [2] "Quadratic Formula" by Wolfram Alpha
  • [3] "Solving Quadratic Equations" by IXL