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 ​

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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, 3x2+24x−24=03x^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, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc 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=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

Applying the Quadratic Formula

Now that we have the quadratic formula, we can apply it to our equation. Plugging in the values of aa, bb, and cc, we get:

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

Simplifying the expression under the square root, we get:

x=−24±576+2886x = \frac{-24 \pm \sqrt{576 + 288}}{6}

x=−24±8646x = \frac{-24 \pm \sqrt{864}}{6}

x=−24±42166x = \frac{-24 \pm 4\sqrt{216}}{6}

x=−24±436⋅66x = \frac{-24 \pm 4\sqrt{36 \cdot 6}}{6}

x=−24±436⋅66x = \frac{-24 \pm 4\sqrt{36} \cdot \sqrt{6}}{6}

x=−24±4⋅6⋅66x = \frac{-24 \pm 4 \cdot 6 \cdot \sqrt{6}}{6}

x=−24±2466x = \frac{-24 \pm 24\sqrt{6}}{6}

x=−246±2466x = \frac{-24}{6} \pm \frac{24\sqrt{6}}{6}

x=−4±46x = -4 \pm 4\sqrt{6}

Conclusion

In this article, we have solved the quadratic equation 3x2+24x−24=03x^2 + 24x - 24 = 0 using the quadratic formula. We have arrived at the solution x=−4±46x = -4 \pm 4\sqrt{6}, which matches option D.

Final Answer

The correct answer is:

  • D. x=−4±26x = -4 \pm 2 \sqrt{6}

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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 our previous article, we solved the quadratic equation 3x2+24x−24=03x^2 + 24x - 24 = 0 using the quadratic formula. In this article, we will address some frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

There are several methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the quadratic formula?

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

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

Q: How do I apply the quadratic formula?

To apply the quadratic formula, you need to plug in the values of aa, bb, and cc into the formula. For example, if you have the quadratic equation x2+5x+6=0x^2 + 5x + 6 = 0, you would plug in a=1a = 1, b=5b = 5, and c=6c = 6 into the formula.

Q: What is the difference between the quadratic formula and factoring?

The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.

Q: Can I use the quadratic formula to solve all quadratic equations?

Yes, the quadratic formula can be used to solve all quadratic equations. However, it may not always be the easiest method to use, especially for equations that can be easily factored.

Q: What is the significance of the discriminant in the quadratic formula?

The discriminant is the expression under the square root in the quadratic formula, b2−4acb^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: Can I use the quadratic formula to solve quadratic equations with complex solutions?

Yes, the quadratic formula can be used to solve quadratic equations with complex solutions. However, the solutions will be in the form of complex numbers.

Conclusion

In this article, we have addressed some frequently asked questions about quadratic equations. We have covered topics such as the definition of a quadratic equation, the quadratic formula, and the significance of the discriminant. We hope that this article has provided you with a better understanding of quadratic equations and how to solve them.

Final Answer

The correct answer is:

  • D. x=−4±26x = -4 \pm 2 \sqrt{6}