Which Equation Shows The Correct Factors For The Quadratic Equation 24 X 2 − 15 = 54 X 24x^2 - 15 = 54x 24 X 2 − 15 = 54 X ?A. 3 X ( 8 X + 5 ) = 54 X 3x(8x + 5) = 54x 3 X ( 8 X + 5 ) = 54 X B. 3 ( 2 X + 5 ) ( 4 X − 1 ) = 0 3(2x + 5)(4x - 1) = 0 3 ( 2 X + 5 ) ( 4 X − 1 ) = 0 C. 3 ( 2 X − 5 ) ( 4 X + 1 ) = 0 3(2x - 5)(4x + 1) = 0 3 ( 2 X − 5 ) ( 4 X + 1 ) = 0 D. 3 ( 8 X 2 − 5 ) = 54 X 3(8x^2 - 5) = 54x 3 ( 8 X 2 − 5 ) = 54 X E. $3(2x - 5)(4x - 3)

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Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields, including algebra, geometry, and calculus. In this article, we will delve into the world of quadratic equations and explore the correct factors for the given equation 24x2−15=54x24x^2 - 15 = 54x. We will examine each option carefully and determine which one shows the correct factors.

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, the quadratic formula, and graphing.

The Given Equation

The given equation is 24x2−15=54x24x^2 - 15 = 54x. To begin solving this equation, we need to isolate the quadratic term by moving all terms to one side of the equation. This gives us 24x2−54x−15=024x^2 - 54x - 15 = 0. Now, we have a quadratic equation in the standard form.

Factoring the Quadratic Equation

To factor the quadratic equation 24x2−54x−15=024x^2 - 54x - 15 = 0, we need to find two numbers whose product is −360-360 (the product of the coefficient of x2x^2 and the constant term) and whose sum is −54-54 (the coefficient of xx). These numbers are −45-45 and 88, as (−45)×8=−360(-45) \times 8 = -360 and (−45)+8=−37(-45) + 8 = -37. However, we need to find two numbers whose sum is −54-54, so we will use −60-60 and 66 instead, as (−60)×6=−360(-60) \times 6 = -360 and (−60)+6=−54(-60) + 6 = -54.

Finding the Correct Factors

Now that we have the numbers −60-60 and 66, we can rewrite the quadratic equation as 24x2−60x+6x−15=024x^2 - 60x + 6x - 15 = 0. We can then factor by grouping: 12x(2x−5)+3(2x−5)=012x(2x - 5) + 3(2x - 5) = 0. Notice that the term (2x−5)(2x - 5) appears in both groups, so we can factor it out: (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0.

Evaluating the Options

Now that we have factored the quadratic equation, we can evaluate the options:

A. 3x(8x+5)=54x3x(8x + 5) = 54x

This option is incorrect because the factored form of the quadratic equation is (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0, not 3x(8x+5)=54x3x(8x + 5) = 54x.

B. 3(2x+5)(4x−1)=03(2x + 5)(4x - 1) = 0

This option is incorrect because the factored form of the quadratic equation is (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0, not 3(2x+5)(4x−1)=03(2x + 5)(4x - 1) = 0.

C. 3(2x−5)(4x+1)=03(2x - 5)(4x + 1) = 0

This option is correct because the factored form of the quadratic equation is (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0, which is equivalent to 3(2x−5)(4x+1)=03(2x - 5)(4x + 1) = 0.

D. 3(8x2−5)=54x3(8x^2 - 5) = 54x

This option is incorrect because the factored form of the quadratic equation is (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0, not 3(8x2−5)=54x3(8x^2 - 5) = 54x.

E. 3(2x−5)(4x−3)=03(2x - 5)(4x - 3) = 0

This option is incorrect because the factored form of the quadratic equation is (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0, not 3(2x−5)(4x−3)=03(2x - 5)(4x - 3) = 0.

Conclusion

In conclusion, the correct factors for the quadratic equation 24x2−15=54x24x^2 - 15 = 54x are (2x−5)(12x+3)=0(2x - 5)(12x + 3) = 0, which is equivalent to option C. 3(2x−5)(4x+1)=03(2x - 5)(4x + 1) = 0. This option shows the correct factors for the given equation.

Final Answer

The final answer is option C. 3(2x−5)(4x+1)=03(2x - 5)(4x + 1) = 0.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields, including algebra, geometry, and calculus. In this article, we will delve into the world of quadratic equations and provide a comprehensive Q&A guide to help you better understand this complex topic.

Q&A: Quadratic Equations

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, 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?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: How do I factor a quadratic equation?

A: Factoring a quadratic equation involves finding two numbers whose product is the constant term and whose sum is the coefficient of the xx term. These numbers are called the factors of the quadratic equation.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation. However, it may not always be the most efficient method, especially for simple equations that can be factored easily.

Q: How do I graph a quadratic equation?

A: Graphing a quadratic equation involves plotting the points on a coordinate plane and drawing a smooth curve through them. The graph of a quadratic equation is a parabola, which is a U-shaped curve.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is the lowest or highest point on the graph, depending on the direction of the parabola.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, many calculators have a built-in quadratic formula function that can be used to solve quadratic equations. However, it's always a good idea to double-check your work and make sure the calculator is set to the correct mode.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields. By following the Q&A guide provided in this article, you should be able to better understand quadratic equations and how to solve them.

Final Tips

  • Always read the problem carefully and make sure you understand what is being asked.
  • Use the quadratic formula or factoring to solve quadratic equations, depending on the specific equation and your personal preference.
  • Graphing a quadratic equation can be a useful way to visualize the solution and understand the behavior of the equation.
  • Use a calculator to check your work and make sure the solution is correct.

Additional Resources

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

By following these tips and resources, you should be able to master quadratic equations and become proficient in solving them.