Which Of The Following Is A Solution Of $x^2 + 5x = -2$?A. 5 ± 33 2 \frac{5 \pm \sqrt{33}}{2} 2 5 ± 33 ​ ​ B. 5 ± 17 2 \frac{5 \pm \sqrt{17}}{2} 2 5 ± 17 ​ ​ C. − 5 ± 33 2 \frac{-5 \pm \sqrt{33}}{2} 2 − 5 ± 33 ​ ​ D. − 5 ± 17 2 \frac{-5 \pm \sqrt{17}}{2} 2 − 5 ± 17 ​ ​

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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, $x^2 + 5x = -2$, and determine which of the given options is a solution to this equation.

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. In our given equation, $x^2 + 5x = -2$, we can rewrite it in the standard form as $x^2 + 5x + 2 = 0$.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our case, a = 1, b = 5, and c = 2. Plugging these values into the quadratic formula, we get:

x=−5±52−4(1)(2)2(1)x = \frac{-5 \pm \sqrt{5^2 - 4(1)(2)}}{2(1)}

Simplifying the expression under the square root, we get:

x=−5±25−82x = \frac{-5 \pm \sqrt{25 - 8}}{2}

x=−5±172x = \frac{-5 \pm \sqrt{17}}{2}

Evaluating the Options

Now that we have the solutions to the equation, let's evaluate the given options:

A. 5±332\frac{5 \pm \sqrt{33}}{2} B. 5±172\frac{5 \pm \sqrt{17}}{2} C. −5±332\frac{-5 \pm \sqrt{33}}{2} D. −5±172\frac{-5 \pm \sqrt{17}}{2}

Comparing the solutions we obtained using the quadratic formula with the given options, we can see that option B matches our solution.

Conclusion

In conclusion, the solution to the quadratic equation $x^2 + 5x = -2$ is $x = \frac{-5 \pm \sqrt{17}}{2}$. Therefore, the correct answer is option B.

Additional Tips and Tricks

  • When solving quadratic equations, it's essential to identify the values of a, b, and c and plug them into the quadratic formula.
  • The quadratic formula can be used to solve quadratic equations with complex solutions.
  • When evaluating the options, make sure to compare the solutions obtained using the quadratic formula with the given options.

Common Mistakes to Avoid

  • Failing to identify the values of a, b, and c and plug them into the quadratic formula.
  • Not simplifying the expression under the square root.
  • Not comparing the solutions obtained using the quadratic formula with the given options.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Final Thoughts

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 address some of the most frequently asked questions about quadratic equations, including their definition, the quadratic formula, and real-world applications.

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 is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of a, b, and c in the equation and plug them into the formula. Then, simplify the expression under the square root and solve for x.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, x) is one. The general form of a linear equation is $ax + b = 0$, where a and b are constants. Quadratic equations, on the other hand, have a degree of two and are in the form $ax^2 + bx + c = 0$.

Q: Can quadratic equations have complex solutions?

A: Yes, quadratic equations can have complex solutions. When the expression under the square root is negative, the solutions will be complex numbers.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

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

A: To determine the number of solutions to a quadratic equation, you need to examine 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: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients. However, you need to be careful when simplifying the expression under the square root.

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

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

  • Failing to identify the values of a, b, and c and plug them into the quadratic formula.
  • Not simplifying the expression under the square root.
  • Not comparing the solutions obtained using the quadratic formula with the given options.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula and how to apply it, you can solve a wide range of quadratic equations and make predictions about real-world systems. Remember to identify the values of a, b, and c, plug them into the quadratic formula, and simplify the expression under the square root. With practice and patience, you'll become a master of solving quadratic equations in no time!