For What Values Of $x$ Is $x^2 + 2x = 24$ True?A. -6 And -4 B. -4 And 6 C. 4 And -6 D. 6 And 4

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 the quadratic equation $x^2 + 2x = 24$ and explore the values of $x$ that make this equation true.

Understanding Quadratic Equations

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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our case, the quadratic equation is $x^2 + 2x = 24$, which can be rewritten as $x^2 + 2x - 24 = 0$.

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:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our case, $a = 1$, $b = 2$, and $c = -24$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-24)}}{2(1)}$$

Simplifying the Quadratic Formula

Simplifying the expression under the square root, we get:

$$x = \frac{-2 \pm \sqrt{4 + 96}}{2}$$

$$x = \frac{-2 \pm \sqrt{100}}{2}$$

$$x = \frac{-2 \pm 10}{2}$$

Finding the Solutions

Now, we can find the two solutions by plugging in the values of $x$:

$$x_1 = \frac{-2 + 10}{2} = \frac{8}{2} = 4$$

$$x_2 = \frac{-2 - 10}{2} = \frac{-12}{2} = -6$$

Conclusion

In this article, we solved the quadratic equation $x^2 + 2x = 24$ using the quadratic formula. We found that the solutions are $x = 4$ and $x = -6$. These values make the equation true, and they are the only solutions to the equation.

Answer

The correct answer is:

• C. 4 and -6

Additional Tips and Resources

• To solve quadratic equations, you can use the quadratic formula or factorization.
• The quadratic formula is a powerful tool for solving quadratic equations, but it can be complex to use.
• Factorization is a simpler method for solving quadratic equations, but it may not always be possible.
• For more information on quadratic equations, check out the following resources:
  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Final Thoughts

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we solved the quadratic equation $x^2 + 2x = 24$ using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional tips and resources for solving them.

Q&A

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 is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

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

• 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:

  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Factorization: Factorization is a simpler method for solving quadratic equations, but it may not always be possible. To factorize a quadratic equation, you need to find two numbers whose product is $ac$ and whose sum is $b$.

• Graphing: Graphing is a visual method for solving quadratic equations. You can graph the quadratic function and find the x-intercepts, which represent the solutions to the equation.

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:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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 under the square root and solve for $x$.

Q: What are the solutions to the quadratic equation $x^2 + 2x = 24$?

A: The solutions to the quadratic equation $x^2 + 2x = 24$ are $x = 4$ and $x = -6$.

Q: How do I check my work?

A: To check your work, you can plug the solutions back into the original equation and verify that they are true. You can also use a calculator or a computer program to check your work.

Additional Tips and Resources

• To solve quadratic equations, you can use the quadratic formula or factorization.
• The quadratic formula is a powerful tool for solving quadratic equations, but it can be complex to use.
• Factorization is a simpler method for solving quadratic equations, but it may not always be possible.
• For more information on quadratic equations, check out the following resources:
  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By using the quadratic formula or factorization, you can find the solutions to quadratic equations and apply them to real-world problems. Remember to always check your work and verify your solutions to ensure accuracy.

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By using the quadratic formula or factorization, you can find the solutions to quadratic equations and apply them to real-world problems. Remember to always check your work and verify your solutions to ensure accuracy.