Given The Equation F ( X ) = 3 X 2 + 6 X + 8 = 0 F(x) = 3x^2 + 6x + 8 = 0 F ( X ) = 3 X 2 + 6 X + 8 = 0 , A) Determine The Axis Of Symmetry And The Vertex.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and other fields. In this article, we will focus on solving the quadratic equation f(x)=3x2+6x+8=0f(x) = 3x^2 + 6x + 8 = 0 and determine the axis of symmetry and the vertex.

What is the Axis of Symmetry?

The axis of symmetry is a line that passes through the vertex of a parabola and is equidistant from the two branches of the parabola. It is a key concept in quadratic equations and is used to determine the vertex of a parabola.

What is the Vertex?

The vertex of a parabola is the highest or lowest point on the parabola. It is the point where the parabola changes direction, and it is the minimum or maximum value of the function.

Solving the Quadratic Equation

To solve the quadratic equation f(x)=3x2+6x+8=0f(x) = 3x^2 + 6x + 8 = 0, we can use the quadratic formula:

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.

In this case, a=3a = 3, b=6b = 6, and c=8c = 8. Plugging these values into the quadratic formula, we get:

x=βˆ’6Β±62βˆ’4(3)(8)2(3)x = \frac{-6 \pm \sqrt{6^2 - 4(3)(8)}}{2(3)}

Simplifying the expression, we get:

x=βˆ’6Β±36βˆ’966x = \frac{-6 \pm \sqrt{36 - 96}}{6}

x=βˆ’6Β±βˆ’606x = \frac{-6 \pm \sqrt{-60}}{6}

x=βˆ’6Β±2i156x = \frac{-6 \pm 2i\sqrt{15}}{6}

x=βˆ’1Β±i153x = -1 \pm \frac{i\sqrt{15}}{3}

Finding the Axis of Symmetry

The axis of symmetry is given by the formula:

x=βˆ’b2ax = -\frac{b}{2a}

In this case, a=3a = 3 and b=6b = 6. Plugging these values into the formula, we get:

x=βˆ’62(3)x = -\frac{6}{2(3)}

x=βˆ’66x = -\frac{6}{6}

x=βˆ’1x = -1

Finding the Vertex

The vertex of a parabola is given by the formula:

(h,k)=(βˆ’b2a,f(βˆ’b2a))(h, k) = (-\frac{b}{2a}, f(-\frac{b}{2a}))

In this case, a=3a = 3, b=6b = 6, and c=8c = 8. Plugging these values into the formula, we get:

(h,k)=(βˆ’62(3),f(βˆ’62(3)))(h, k) = (-\frac{6}{2(3)}, f(-\frac{6}{2(3)}))

(h,k)=(βˆ’1,f(βˆ’1))(h, k) = (-1, f(-1))

To find the value of f(βˆ’1)f(-1), we plug x=βˆ’1x = -1 into the original equation:

f(βˆ’1)=3(βˆ’1)2+6(βˆ’1)+8f(-1) = 3(-1)^2 + 6(-1) + 8

f(βˆ’1)=3βˆ’6+8f(-1) = 3 - 6 + 8

f(βˆ’1)=5f(-1) = 5

Therefore, the vertex of the parabola is (βˆ’1,5)(-1, 5).

Conclusion

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and other fields. In this article, we will provide a Q&A guide to help you better understand quadratic equations and how to solve them.

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. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

  • Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve for the variable by setting each factor equal to zero.
  • Quadratic formula: The quadratic formula is a general method for solving quadratic equations. It is given by the formula x = (-b Β± √(b^2 - 4ac)) / 2a.
  • Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a general method for solving quadratic equations. It is given by the formula x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression and solve for x.

Q: What is the axis of symmetry?

A: The axis of symmetry is a line that passes through the vertex of a parabola and is equidistant from the two branches of the parabola. It is a key concept in quadratic equations and is used to determine the vertex of a parabola.

Q: How do I find the axis of symmetry?

A: To find the axis of symmetry, you can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic equation.

Q: What is the vertex?

A: The vertex of a parabola is the highest or lowest point on the parabola. It is the point where the parabola changes direction, and it is the minimum or maximum value of the function.

Q: How do I find the vertex?

A: To find the vertex, you can use the formula (h, k) = (-b / 2a, f(-b / 2a)), where a, b, and c are the coefficients of the quadratic equation.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

  • Not simplifying the expression correctly
  • Not using the correct formula
  • Not checking the solutions for extraneous solutions
  • Not graphing the related function to check the solutions

Conclusion

In this article, we provided a Q&A guide to help you better understand quadratic equations and how to solve them. We covered topics such as the quadratic formula, axis of symmetry, and vertex, and provided examples and explanations to help you understand the concepts. By following the tips and avoiding common mistakes, you can become more confident and proficient in solving quadratic equations.

Additional Resources

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

Practice Problems

  • Solve the quadratic equation x^2 + 5x + 6 = 0 using the quadratic formula.
  • Find the axis of symmetry and vertex of the parabola given by the equation x^2 + 4x + 4 = 0.
  • Solve the quadratic equation x^2 - 7x + 12 = 0 using factoring.

Answer Key

  • x = (-5 Β± √(25 - 24)) / 2
  • x = -2
  • x = 3 or x = 4