Solve For $x$ In The Equation $2x^2 + 3x - 7 = X^2 + 5x + 39$.A. $ X = − 6 ± 82 X = -6 \pm \sqrt{82} X = − 6 ± 82 ​ [/tex]B. $x = -6 \pm 2\sqrt{17}$C. $x = 1 + \sqrt{33}$D. $ X = 1 ± 47 X = 1 \pm \sqrt{47} X = 1 ± 47 ​ [/tex]

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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 a specific quadratic equation, $2x^2 + 3x - 7 = x^2 + 5x + 39$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Given Equation

The given equation is $2x^2 + 3x - 7 = x^2 + 5x + 39$. To solve for x, we need to isolate the variable on one side of the equation. We can start by combining like terms and simplifying the equation.

Step 1: Combine Like Terms

First, we will combine the like terms on both sides of the equation.

2x2+3x−7=x2+5x+392x^2 + 3x - 7 = x^2 + 5x + 39

Subtracting $x^2 + 5x + 39$ from both sides gives us:

x2−2x−46=0x^2 - 2x - 46 = 0

Step 2: Simplify the Equation

Now, we can simplify the equation by combining the like terms.

x2−2x−46=0x^2 - 2x - 46 = 0

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where a = 1, b = -2, and c = -46.

Using the Quadratic Formula

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

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

In our case, a = 1, b = -2, and c = -46. Plugging these values into the quadratic formula, we get:

x=−(−2)±(−2)2−4(1)(−46)2(1)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-46)}}{2(1)}

Simplifying the expression, we get:

x=2±4+1842x = \frac{2 \pm \sqrt{4 + 184}}{2}

x=2±1882x = \frac{2 \pm \sqrt{188}}{2}

x=2±2472x = \frac{2 \pm 2\sqrt{47}}{2}

x=1±47x = 1 \pm \sqrt{47}

Conclusion

In this article, we solved the quadratic equation $2x^2 + 3x - 7 = x^2 + 5x + 39$ using the quadratic formula. We started by combining like terms and simplifying the equation, and then used the quadratic formula to find the solutions. The solutions to the equation are $x = 1 \pm \sqrt{47}$.

Discussion

The quadratic formula is a powerful tool for solving quadratic equations. However, it can be challenging to apply, especially when dealing with complex numbers. In this case, we used the quadratic formula to find the solutions to the equation, but we could have also used other methods, such as factoring or graphing.

Final Answer

The final answer to the equation $2x^2 + 3x - 7 = x^2 + 5x + 39$ is:

x=1±47x = 1 \pm \sqrt{47}

This is the correct answer, and it matches option D in the given choices.

Additional Resources

For more information on quadratic equations and the quadratic formula, check out the following resources:

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

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 answer some of the most frequently asked questions about quadratic equations, including their definition, properties, and methods for solving 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 (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.

Q: What are the Properties of Quadratic Equations?

A: Quadratic equations have several properties, including:

  • Real and Complex Roots: Quadratic equations can have real or complex roots, depending on the value of the discriminant (b^2 - 4ac).
  • Symmetry: Quadratic equations are symmetric about the axis of symmetry, which is given by the formula x = -b/2a.
  • Vertex Form: Quadratic equations can be written in vertex form, which is given by the formula f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods for solving quadratic equations, including:

  • Factoring: Factoring involves expressing the quadratic equation as a product of two binomials.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by the formula x = (-b ± √(b^2 - 4ac)) / 2a.
  • Graphing: Graphing involves plotting the quadratic equation on a coordinate plane and finding the x-intercepts.

Q: What is the Discriminant?

A: The discriminant is a value that is used to determine the nature of the roots of a quadratic equation. It is given by the formula b^2 - 4ac. If the discriminant is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.

Q: What is the Axis of Symmetry?

A: The axis of symmetry is a line that passes through the vertex of the parabola and is perpendicular to the x-axis. It is given by the formula x = -b/2a.

Q: What is the Vertex Form of a Quadratic Equation?

A: The vertex form of a quadratic equation is given by the formula f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

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 plot the equation on a coordinate plane and find the x-intercepts.

Q: What are Some Common Quadratic Equations?

A: Some common quadratic equations include:

  • x^2 + 4x + 4 = 0: This equation has two real roots, x = -2.
  • x^2 - 6x + 8 = 0: This equation has two real roots, x = 2 and x = 4.
  • x^2 + 2x + 1 = 0: This equation has one real root, x = -1.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we answered some of the most frequently asked questions about quadratic equations, including their definition, properties, and methods for solving them. We hope this article has been helpful in understanding quadratic equations and their applications.

Additional Resources

For more information on quadratic equations and their applications, check out the following resources:

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

These resources provide a comprehensive overview of quadratic equations and their applications, and they offer step-by-step examples and solutions to help you understand the material.