What Are The Solutions Of This Quadratic Equation?$x^2 - 6x + 2 = 0$A. $x = 6 \pm \sqrt{7}$B. $x = 3 \pm \sqrt{7}$C. $x = -3 \pm \sqrt{7}$D. $x = 3 \pm \sqrt{11}$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. 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, and xx is the variable. In this article, we will focus on solving a specific quadratic equation, x2−6x+2=0x^2 - 6x + 2 = 0, and explore the different solutions that can be obtained.

The Quadratic Formula

To solve a quadratic equation, we can use the quadratic formula, which is given by:

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

This formula provides two solutions for the quadratic equation, which are given by the plus and minus signs. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of any quadratic equation.

Applying the Quadratic Formula to the Given Equation

Now, let's apply the quadratic formula to the given equation, x2−6x+2=0x^2 - 6x + 2 = 0. In this equation, a=1a = 1, b=−6b = -6, and c=2c = 2. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

x=6±36−82x = \frac{6 \pm \sqrt{36 - 8}}{2}

x=6±282x = \frac{6 \pm \sqrt{28}}{2}

x=6±4×72x = \frac{6 \pm \sqrt{4 \times 7}}{2}

x=6±272x = \frac{6 \pm 2\sqrt{7}}{2}

x=3±7x = 3 \pm \sqrt{7}

Evaluating the Solutions

Now that we have obtained the solutions using the quadratic formula, let's evaluate them. The solutions are given by x=3±7x = 3 \pm \sqrt{7}. This means that the solutions are two values of xx that satisfy the equation x2−6x+2=0x^2 - 6x + 2 = 0. The plus sign gives us the value of xx that is greater than 3, while the minus sign gives us the value of xx that is less than 3.

Comparing the Solutions with the Options

Now, let's compare the solutions we obtained with the options given in the problem. The options are:

A. x=6±7x = 6 \pm \sqrt{7} B. x=3±7x = 3 \pm \sqrt{7} C. x=−3±7x = -3 \pm \sqrt{7} D. x=3±11x = 3 \pm \sqrt{11}

Comparing the solutions we obtained with the options, we can see that the correct solution is option B, which is x=3±7x = 3 \pm \sqrt{7}.

Conclusion

In this article, we solved a quadratic equation using the quadratic formula. We obtained the solutions x=3±7x = 3 \pm \sqrt{7}, and we compared them with the options given in the problem. We found that the correct solution is option B, which is x=3±7x = 3 \pm \sqrt{7}. This demonstrates the power of the quadratic formula in solving quadratic equations and provides a clear understanding of the solutions that can be obtained.

Frequently Asked Questions

  • What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations, and it is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • How do I apply the quadratic formula to a quadratic equation? To apply the quadratic formula to a quadratic equation, you need to plug in the values of aa, bb, and cc into the formula and simplify the expression.
  • What are the solutions of the quadratic equation x2−6x+2=0x^2 - 6x + 2 = 0? The solutions of the quadratic equation x2−6x+2=0x^2 - 6x + 2 = 0 are x=3±7x = 3 \pm \sqrt{7}.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of any quadratic equation. In this article, we solved a quadratic equation using the quadratic formula and obtained the solutions x=3±7x = 3 \pm \sqrt{7}. We compared the solutions with the options given in the problem and found that the correct solution is option B, which is x=3±7x = 3 \pm \sqrt{7}. This demonstrates the power of the quadratic formula in solving quadratic equations and provides a clear understanding of the solutions that can be obtained.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. In our previous article, we solved a quadratic equation using the quadratic formula and obtained the solutions x=3±7x = 3 \pm \sqrt{7}. In this article, we will provide a comprehensive Q&A guide to help you understand the solutions of quadratic equations and how to apply the quadratic formula.

Q&A Guide

Q1: What is the quadratic formula?

A1: The quadratic formula is a powerful tool for solving quadratic equations, and it is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q2: How do I apply the quadratic formula to a quadratic equation?

A2: To apply the quadratic formula to a quadratic equation, you need to plug in the values of aa, bb, and cc into the formula and simplify the expression.

Q3: What are the solutions of the quadratic equation x2−6x+2=0x^2 - 6x + 2 = 0?

A3: The solutions of the quadratic equation x2−6x+2=0x^2 - 6x + 2 = 0 are x=3±7x = 3 \pm \sqrt{7}.

Q4: What is the significance of the quadratic formula?

A4: The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of any quadratic equation. It is a fundamental concept in mathematics and has numerous applications in various fields.

Q5: How do I determine the number of solutions of a quadratic equation?

A5: To determine the number of solutions of a quadratic equation, you need to check the discriminant, which is given by b2−4acb^2 - 4ac. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q6: What is the difference between a quadratic equation and a linear equation?

A6: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one.

Q7: How do I graph a quadratic equation?

A7: To graph a quadratic equation, you need to find the x-intercepts and the vertex of the parabola. The x-intercepts are the points where the parabola intersects the x-axis, and the vertex is the point where the parabola is at its maximum or minimum value.

Q8: What is the vertex form of a quadratic equation?

A8: The vertex form of a quadratic equation is given by y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.

Q9: How do I convert a quadratic equation from standard form to vertex form?

A9: To convert a quadratic equation from standard form to vertex form, you need to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q10: What is the significance of the axis of symmetry in a quadratic equation?

A10: The axis of symmetry is a line that passes through the vertex of the parabola and is perpendicular to the x-axis. It is a fundamental concept in mathematics and has numerous applications in various fields.

Conclusion

In this article, we provided a comprehensive Q&A guide to help you understand the solutions of quadratic equations and how to apply the quadratic formula. We covered topics such as the quadratic formula, the significance of the quadratic formula, and how to determine the number of solutions of a quadratic equation. We also covered topics such as graphing quadratic equations, the vertex form of a quadratic equation, and the significance of the axis of symmetry. We hope that this article has been helpful in providing you with a deeper understanding of quadratic equations and how to apply the quadratic formula.

Frequently Asked Questions

  • What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations, and it is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • How do I apply the quadratic formula to a quadratic equation? To apply the quadratic formula to a quadratic equation, you need to plug in the values of aa, bb, and cc into the formula and simplify the expression.
  • What are the solutions of the quadratic equation x2−6x+2=0x^2 - 6x + 2 = 0? The solutions of the quadratic equation x2−6x+2=0x^2 - 6x + 2 = 0 are x=3±7x = 3 \pm \sqrt{7}.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of any quadratic equation. In this article, we provided a comprehensive Q&A guide to help you understand the solutions of quadratic equations and how to apply the quadratic formula. We hope that this article has been helpful in providing you with a deeper understanding of quadratic equations and how to apply the quadratic formula.