Which Values Of { X $}$ Are Solutions To The Equation Below? Check All That Apply.${ 4x^2 - 30 = 34 }$A. { X = \sqrt{8} $}$ B. { X = -8 $}$ C. { X = -\sqrt{8} $}$ D. { X = 4 $}$ E.

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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 explore the process of solving quadratic equations, with a focus on the given equation: 4x2−30=344x^2 - 30 = 34. We will examine the possible values of xx that satisfy this equation and discuss the steps involved in solving quadratic equations.

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 ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. In the given equation, 4x2−30=344x^2 - 30 = 34, we can rewrite it in the standard form as 4x2−64=04x^2 - 64 = 0 by subtracting 34 from both sides.

Solving the Equation

To solve the equation 4x2−64=04x^2 - 64 = 0, we can use the method of factoring or the quadratic formula. In this case, we will use the quadratic formula, which is given by:

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

In our equation, a=4a = 4, b=0b = 0, and c=−64c = -64. Plugging these values into the quadratic formula, we get:

x=−0±02−4(4)(−64)2(4)x = \frac{-0 \pm \sqrt{0^2 - 4(4)(-64)}}{2(4)}

Simplifying the expression under the square root, we get:

x=±10248x = \frac{\pm \sqrt{1024}}{8}

x=±328x = \frac{\pm 32}{8}

x=±4x = \pm 4

Evaluating the Solutions

Now that we have found the possible values of xx, we need to evaluate them to see if they satisfy the original equation. Let's substitute x=4x = 4 and x=−4x = -4 into the original equation to see if they are true:

For x=4x = 4:

4(4)2−30=64−30=344(4)^2 - 30 = 64 - 30 = 34

This is true, so x=4x = 4 is a solution to the equation.

For x=−4x = -4:

4(−4)2−30=64−30=344(-4)^2 - 30 = 64 - 30 = 34

This is also true, so x=−4x = -4 is a solution to the equation.

Conclusion

In this article, we have solved the quadratic equation 4x2−30=344x^2 - 30 = 34 using the quadratic formula. We have found that the possible values of xx are x=4x = 4 and x=−4x = -4. These values satisfy the original equation, and we have verified them by substituting them back into the equation.

Discussion

Now that we have solved the equation, let's discuss the possible values of xx that are given in the options:

A. x=8x = \sqrt{8}

This is not a solution to the equation, as we have found that x=4x = 4 and x=−4x = -4 are the only solutions.

B. x=−8x = -8

This is not a solution to the equation, as we have found that x=4x = 4 and x=−4x = -4 are the only solutions.

C. x=−8x = -\sqrt{8}

This is not a solution to the equation, as we have found that x=4x = 4 and x=−4x = -4 are the only solutions.

D. x=4x = 4

This is a solution to the equation, as we have verified it by substituting it back into the equation.

E. (No option given)

Final Answer

Based on our analysis, the correct answers are:

  • D. x=4x = 4
  • E. (No option given, but x=−4x = -4 is also a solution)

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, providing clear and concise answers to help you better understand this important 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 is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method, 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 provides the solutions to a quadratic equation. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of aa, bb, and cc in the quadratic equation. Then, plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the possible solutions to a quadratic equation?

A: The possible solutions to a quadratic equation are the values of xx that satisfy the equation. These solutions can be real or complex numbers.

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

A: The number of solutions to a quadratic equation depends on the discriminant, which is the expression under the square root in the quadratic formula. 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.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by:

b2−4acb^2 - 4ac

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot the graph.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is given by the formula:

x=−b2ax = -\frac{b}{2a}

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you can use the formula:

x=−b2ax = -\frac{b}{2a}

Q: What is the axis of symmetry of a quadratic equation?

A: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the parabola. It is given by the formula:

x=−b2ax = -\frac{b}{2a}

Q: How do I find the axis of symmetry of a quadratic equation?

A: To find the axis of symmetry of a quadratic equation, you can use the formula:

x=−b2ax = -\frac{b}{2a}

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important topic. Whether you are a student or a professional, quadratic equations are an essential part of mathematics, and mastering them will help you solve a wide range of problems in various fields.

Additional Resources

For more information on quadratic equations, you can consult the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebra and geometry. By mastering quadratic equations, you will be able to solve a wide range of problems in various fields, from science and engineering to economics and finance.