Solve For $x$. Write Both Solutions, Separated By A Comma.$6x^2 + 7x + 2 = 0$

by ADMIN 82 views

===========================================================

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 delve into the world of quadratic equations and provide a step-by-step guide on how to solve them. We will focus on the quadratic equation 6x2+7x+2=06x^2 + 7x + 2 = 0 and provide both solutions, separated by a comma.

What are Quadratic Equations?

Quadratic equations are a type of polynomial equation that contains a squared variable, usually xx. They are often written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Quadratic Formula

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

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation. To use the quadratic formula, we need to plug in the values of aa, bb, and cc into the formula and simplify.

Solving the Quadratic Equation 6x2+7x+2=06x^2 + 7x + 2 = 0

Now that we have introduced the quadratic formula, let's apply it to the quadratic equation 6x2+7x+2=06x^2 + 7x + 2 = 0. We can plug in the values of aa, bb, and cc into the quadratic formula:

x=βˆ’7Β±72βˆ’4(6)(2)2(6)x = \frac{-7 \pm \sqrt{7^2 - 4(6)(2)}}{2(6)}

Simplifying the expression under the square root, we get:

x=βˆ’7Β±49βˆ’4812x = \frac{-7 \pm \sqrt{49 - 48}}{12}

x=βˆ’7Β±112x = \frac{-7 \pm \sqrt{1}}{12}

x=βˆ’7Β±112x = \frac{-7 \pm 1}{12}

Solution 1

The first solution is obtained by using the plus sign in the quadratic formula:

x=βˆ’7+112x = \frac{-7 + 1}{12}

x=βˆ’612x = \frac{-6}{12}

x=βˆ’12x = -\frac{1}{2}

Solution 2

The second solution is obtained by using the minus sign in the quadratic formula:

x=βˆ’7βˆ’112x = \frac{-7 - 1}{12}

x=βˆ’812x = \frac{-8}{12}

x=βˆ’23x = -\frac{2}{3}

Conclusion

In this article, we have provided a comprehensive guide on how to solve quadratic equations using the quadratic formula. We have applied the quadratic formula to the quadratic equation 6x2+7x+2=06x^2 + 7x + 2 = 0 and obtained two solutions, separated by a comma. The first solution is x=βˆ’12x = -\frac{1}{2}, and the second solution is x=βˆ’23x = -\frac{2}{3}. We hope that this article has provided valuable insights and skills for students and professionals alike.

Frequently Asked Questions

Q: What is the quadratic formula?

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

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

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

Q: What are the two solutions to the quadratic equation 6x2+7x+2=06x^2 + 7x + 2 = 0?

A: The two solutions are x=βˆ’12x = -\frac{1}{2} and x=βˆ’23x = -\frac{2}{3}.

References

Further Reading

================================================================

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 provide a comprehensive Q&A guide on quadratic equations, covering various topics and concepts. Whether you are a student, teacher, or professional, this article will provide valuable insights and answers to your questions.

Q: What is a quadratic equation?

A: A quadratic equation is a type of polynomial equation that contains a squared variable, usually xx. It is often written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: What is the quadratic formula?

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

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

A: To apply the quadratic formula, you need to plug in the values of aa, bb, and cc into the formula and simplify. Make sure to follow the order of operations (PEMDAS) and simplify the expression under the square root.

Q: What are the two solutions to the quadratic equation 6x2+7x+2=06x^2 + 7x + 2 = 0?

A: The two solutions are x=βˆ’12x = -\frac{1}{2} and x=βˆ’23x = -\frac{2}{3}.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring. However, not all quadratic equations can be factored easily. The quadratic formula is a more general method for solving quadratic equations.

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

A: A quadratic equation is a type of polynomial equation that contains a squared variable, while a linear equation is a type of polynomial equation that contains only a first-degree variable.

Q: Can I use the quadratic formula to solve a cubic equation?

A: No, the quadratic formula is only applicable to quadratic equations. To solve a cubic equation, you need to use a different method, such as the cubic formula.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, given by b2βˆ’4acb^2 - 4ac. If the discriminant is positive, the quadratic equation has two distinct real solutions. If the discriminant is zero, the quadratic equation has one real solution. If the discriminant is negative, the quadratic equation has no real solutions.

Q: Can I use the quadratic formula to solve a quadratic equation with complex coefficients?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex coefficients. However, the solutions will be complex numbers.

Q: What is the relationship between the quadratic formula and the graph of a quadratic function?

A: The quadratic formula is related to the graph of a quadratic function. The solutions to the quadratic equation correspond to the x-intercepts of the graph of the quadratic function.

Q: Can I use the quadratic formula to solve a quadratic equation with a variable coefficient?

A: No, the quadratic formula is only applicable to quadratic equations with constant coefficients. To solve a quadratic equation with a variable coefficient, you need to use a different method.

Conclusion

In this article, we have provided a comprehensive Q&A guide on quadratic equations, covering various topics and concepts. Whether you are a student, teacher, or professional, this article will provide valuable insights and answers to your questions. We hope that this article has helped you to better understand quadratic equations and how to solve them using the quadratic formula.

Frequently Asked Questions

Q: What is the quadratic formula?

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

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

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

Q: What are the two solutions to the quadratic equation 6x2+7x+2=06x^2 + 7x + 2 = 0?

A: The two solutions are x=βˆ’12x = -\frac{1}{2} and x=βˆ’23x = -\frac{2}{3}.

References

Further Reading