Select The Correct Answer.Solve The Equation: 3 X 2 + 24 X − 24 = 0 3x^2 + 24x - 24 = 0 3 X 2 + 24 X − 24 = 0 A. X = 2 ± 4 6 X = 2 \pm 4\sqrt{6} X = 2 ± 4 6 B. X = − 2 ± 4 6 X = -2 \pm 4\sqrt{6} X = − 2 ± 4 6 C. X = 4 ± 2 6 X = 4 \pm 2\sqrt{6} X = 4 ± 2 6 D. X = − 4 ± 2 6 X = -4 \pm 2\sqrt{6} X = − 4 ± 2 6

Mar 11, 2025 by ADMIN 316 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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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, $3x^2 + 24x - 24 = 0$ , and provide a step-by-step guide on how to arrive at the correct solution.

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, completing the square, and the quadratic formula.

The Quadratic Formula

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

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation. In our case, $a = 3$ , $b = 24$ , and $c = -24$ .

Applying the Quadratic Formula

To solve the equation $3x^2 + 24x - 24 = 0$ , we will substitute the values of $a$ , $b$ , and $c$ into the quadratic formula:

$x = \frac{-24 \pm \sqrt{(24)^2 - 4(3)(-24)}}{2(3)}$

Simplifying the expression under the square root, we get:

$x = \frac{-24 \pm \sqrt{576 + 288}}{6}$

$x = \frac{-24 \pm \sqrt{864}}{6}$

$x = \frac{-24 \pm 12\sqrt{6}}{6}$

Simplifying the Solution

To simplify the solution, we can divide both the numerator and the denominator by 6:

$x = \frac{-4 \pm 2\sqrt{6}}{1}$

$x = -4 \pm 2\sqrt{6}$

Conclusion

In this article, we have solved the quadratic equation $3x^2 + 24x - 24 = 0$ using the quadratic formula. We have arrived at the solution $x = -4 \pm 2\sqrt{6}$ , which is the correct answer.

Comparison with Answer Choices

Let's compare our solution with the answer choices:

A. $x = 2 \pm 4\sqrt{6}$ : This is not the correct solution.
B. $x = -2 \pm 4\sqrt{6}$ : This is not the correct solution.
C. $x = 4 \pm 2\sqrt{6}$ : This is not the correct solution.
D. $x = -4 \pm 2\sqrt{6}$ : This is the correct solution.

Final Answer

The correct answer is:

D. $x = -4 \pm 2\sqrt{6}$

Tips and Tricks

When solving quadratic equations, make sure to identify the coefficients $a$ , $b$ , and $c$ correctly.
Use the quadratic formula when the equation cannot be factored easily.
Simplify the solution by dividing both the numerator and the denominator by the greatest common factor.

Practice Problems

Try solving the following quadratic equations using the quadratic formula:

$x^2 + 5x + 6 = 0$
$2x^2 - 3x - 1 = 0$
$x^2 + 2x - 15 = 0$

Conclusion

Solving quadratic equations is an essential skill in mathematics. By using the quadratic formula and simplifying the solution, we can arrive at the correct answer. Remember to identify the coefficients correctly, use the quadratic formula when necessary, and simplify the solution to get the final answer.

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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 provide a Q&A guide on quadratic equations, covering various topics and concepts.

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 different methods for solving quadratic equations?

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

Factoring: This method involves expressing the quadratic equation as a product of two binomials.
Completing the square: This method involves rewriting the quadratic equation in a form that allows us to easily find the solutions.
Quadratic formula: This method involves using a formula to find the solutions of the quadratic equation.

Q: What is the quadratic formula?

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

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to substitute 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 steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

Identify the coefficients $a$ , $b$ , and $c$ of the quadratic equation.
Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula.
Simplify the expression under the square root.
Solve for $x$ .

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

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

Not identifying the coefficients $a$ , $b$ , and $c$ correctly.
Not simplifying the expression under the square root.
Not solving for $x$ correctly.

Q: How do I check my solutions?

A: To check your solutions, you can substitute the values of $x$ back into the original quadratic equation and verify that the equation is true.

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

A: Quadratic equations have many 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: Can you provide some practice problems for solving quadratic equations?

A: Yes, here are some practice problems for solving quadratic equations:

$x^2 + 5x + 6 = 0$
$2x^2 - 3x - 1 = 0$
$x^2 + 2x - 15 = 0$

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the different methods for solving quadratic equations, applying the quadratic formula, and checking your solutions, you can become proficient in solving quadratic equations. Remember to avoid common mistakes and practice regularly to improve your skills.