Which Of The Following Are Solutions To The Equation Below? Check All That Apply. 3 X 2 + 27 X + 54 = 0 3x^2 + 27x + 54 = 0 3 X 2 + 27 X + 54 = 0 A. 9 B. -3 C. -6 D. 3 E. 6

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 explore the solutions to the quadratic equation $3x^2 + 27x + 54 = 0$. We will examine each option and determine which ones are valid solutions.

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 equation, $a = 3$, $b = 27$, and $c = 54$.

The Quadratic Formula

To solve quadratic equations, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula provides two solutions for the equation, which are the values of $x$ that satisfy the equation.

Applying the Quadratic Formula

Let's apply the quadratic formula to our equation: $3x^2 + 27x + 54 = 0$. We have $a = 3$, $b = 27$, and $c = 54$. Plugging these values into the quadratic formula, we get:

$x = \frac{-27 \pm \sqrt{27^2 - 4(3)(54)}}{2(3)}$

$x = \frac{-27 \pm \sqrt{729 - 648}}{6}$

$x = \frac{-27 \pm \sqrt{81}}{6}$

$x = \frac{-27 \pm 9}{6}$

This gives us two possible solutions:

$x = \frac{-27 + 9}{6}$ or $x = \frac{-27 - 9}{6}$

$x = \frac{-18}{6}$ or $x = \frac{-36}{6}$

$x = -3$ or $x = -6$

Evaluating the Options

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

A. 9: This is not a solution to the equation.

B. -3: This is a solution to the equation.

C. -6: This is a solution to the equation.

D. 3: This is not a solution to the equation.

E. 6: This is not a solution to the equation.

Conclusion

In conclusion, the solutions to the quadratic equation $3x^2 + 27x + 54 = 0$ are $x = -3$ and $x = -6$. Therefore, the correct options are B and C.

Additional Tips and Resources

• To solve quadratic equations, you can use the quadratic formula or factorization.
• Make sure to check your solutions by plugging them back into the original equation.
• For more practice, try solving quadratic equations with different coefficients and constants.
• You can also use online resources, such as Khan Academy or Wolfram Alpha, to help you solve quadratic equations.

Final Thoughts

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important topic.

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 to solve quadratic equations, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve the equation by setting each factor equal to zero.
• Quadratic formula: The quadratic formula is a general method for solving quadratic equations: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Graphing: You can also solve quadratic equations 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: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula provides two solutions for the equation, which are the values of $x$ that satisfy the equation.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. For example, if you have the equation $x^2 + 5x + 6 = 0$, you would plug in $a = 1$, $b = 5$, and $c = 6$ into the formula.

Q: What is the discriminant?

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

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. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can I use the quadratic formula to solve equations with complex solutions?

A: Yes, you can use the quadratic formula to solve equations with complex solutions. If the discriminant is negative, the quadratic formula will provide two complex solutions.

Q: How do I check my solutions to a quadratic equation?

A: To check your solutions to a quadratic equation, you need to plug the solutions back into the original equation. If the solutions satisfy the equation, they are valid solutions.

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

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

• Not checking the solutions to the equation.
• Not using the correct values of $a$, $b$, and $c$ in the quadratic formula.
• Not simplifying the expression under the square root in the quadratic formula.
• Not considering complex solutions to the equation.

Conclusion

In conclusion, quadratic equations are an essential part of mathematics, and solving them requires practice and patience. By understanding the quadratic formula and the discriminant, you can solve quadratic equations with confidence. Remember to check your solutions and use online resources to help you practice. With time and practice, you will become proficient in solving quadratic equations and tackle more complex mathematical problems.