A Student Is Deriving The Quadratic Formula. Her First Two Steps Are Shown:Step 1: $ -c = A X^2 + B X $ Step 2: $ -c = A \left(x^2 + \frac{b}{a} X\right) $ Which Best Explains Or Justifies Step 2?A. Division Property Of Equality B.

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Introduction

The quadratic formula is a fundamental concept in algebra, used to solve quadratic equations of the form ax^2 + bx + c = 0. It is derived by manipulating the given equation to isolate the variable x. In this article, we will focus on the second step of deriving the quadratic formula, where the equation is rewritten to facilitate further manipulation.

Step 1: The Given Equation

The first step in deriving the quadratic formula is to write the given equation in the standard form:

−c=ax2+bx-c = ax^2 + bx

This equation represents a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

Step 2: Factoring Out the Coefficient of x^2

The second step in deriving the quadratic formula involves factoring out the coefficient of x^2, which is a. This is done by multiplying both sides of the equation by a, resulting in:

−c=a(x2+bax)-c = a\left(x^2 + \frac{b}{a} x\right)

This step is crucial in deriving the quadratic formula, as it allows us to rewrite the equation in a form that is easier to manipulate.

Justification for Step 2

So, which property of equality justifies Step 2? The correct answer is the Multiplication Property of Equality. This property states that if we multiply both sides of an equation by the same non-zero value, the resulting equation is true if and only if the original equation is true.

In this case, we multiplied both sides of the equation by a, which is a non-zero value. This operation is valid, and the resulting equation is true if and only if the original equation is true.

Alternative Justification

Some students may argue that the correct answer is the Division Property of Equality. However, this is not the case. The Division Property of Equality states that if we divide both sides of an equation by the same non-zero value, the resulting equation is true if and only if the original equation is true.

In this case, we multiplied both sides of the equation by a, not divided. Therefore, the Division Property of Equality does not apply.

Conclusion

In conclusion, the second step in deriving the quadratic formula involves factoring out the coefficient of x^2, which is a. This is done by multiplying both sides of the equation by a, resulting in:

−c=a(x2+bax)-c = a\left(x^2 + \frac{b}{a} x\right)

The justification for this step is the Multiplication Property of Equality, which states that if we multiply both sides of an equation by the same non-zero value, the resulting equation is true if and only if the original equation is true.

Deriving the Quadratic Formula: The Final Steps

Now that we have derived the quadratic formula up to the second step, we can proceed to the final steps. In the next article, we will discuss the third step, where we complete the square to isolate the variable x.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "College Algebra" by James Stewart

Further Reading

  • "Deriving the Quadratic Formula: The Third Step"
  • "Solving Quadratic Equations: A Step-by-Step Guide"

FAQs

  • Q: What is the quadratic formula? A: The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0.
  • Q: What is the second step in deriving the quadratic formula? A: The second step involves factoring out the coefficient of x^2, which is a.
  • Q: What property of equality justifies the second step? A: The Multiplication Property of Equality justifies the second step.
    Deriving the Quadratic Formula: A Q&A Guide =====================================================

Introduction

In our previous article, we discussed the first two steps in deriving the quadratic formula. We also justified the second step using the Multiplication Property of Equality. In this article, we will continue to answer some of the most frequently asked questions about deriving the quadratic formula.

Q&A Guide

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0. It is given by:

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

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It allows us to find the solutions to quadratic equations in a straightforward and efficient manner.

Q: What are the steps involved in deriving the quadratic formula?

A: The steps involved in deriving the quadratic formula are:

  1. Write the given equation in the standard form.
  2. Factor out the coefficient of x^2.
  3. Complete the square to isolate the variable x.
  4. Solve for x.

Q: What is the justification for the second step in deriving the quadratic formula?

A: The justification for the second step is the Multiplication Property of Equality. This property states that if we multiply both sides of an equation by the same non-zero value, the resulting equation is true if and only if the original equation is true.

Q: What is the significance of the coefficient of x^2 in the quadratic formula?

A: The coefficient of x^2 is the value of a in the quadratic equation. It plays a crucial role in determining the solutions to the quadratic equation.

Q: How do we complete the square to isolate the variable x?

A: To complete the square, we add and subtract the square of half the coefficient of x to the equation. This allows us to rewrite the equation in a form that is easier to solve.

Q: What is the final step in deriving the quadratic formula?

A: The final step involves solving for x. This is done by simplifying the equation and isolating the variable x.

Q: What are some common mistakes to avoid when deriving the quadratic formula?

A: Some common mistakes to avoid when deriving the quadratic formula include:

  • Not factoring out the coefficient of x^2.
  • Not completing the square correctly.
  • Not solving for x correctly.

Q: How do we apply the quadratic formula to solve quadratic equations?

A: To apply the quadratic formula, we substitute the values of a, b, and c into the formula and simplify. This gives us the solutions to the quadratic equation.

Conclusion

In conclusion, deriving the quadratic formula is a straightforward process that involves several steps. By following these steps and understanding the justification for each step, we can derive the quadratic formula and apply it to solve quadratic equations.

Deriving the Quadratic Formula: A Step-by-Step Guide

If you are interested in learning more about deriving the quadratic formula, we recommend checking out our previous article, "Deriving the Quadratic Formula: The First Two Steps". This article provides a step-by-step guide to deriving the quadratic formula and justifies each step using the Multiplication Property of Equality.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "College Algebra" by James Stewart

Further Reading

  • "Deriving the Quadratic Formula: The Third Step"
  • "Solving Quadratic Equations: A Step-by-Step Guide"

FAQs

  • Q: What is the quadratic formula? A: The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0.
  • Q: What is the significance of the quadratic formula? A: The quadratic formula is a powerful tool for solving quadratic equations.
  • Q: What are the steps involved in deriving the quadratic formula? A: The steps involved in deriving the quadratic formula are:
    1. Write the given equation in the standard form.
    2. Factor out the coefficient of x^2.
    3. Complete the square to isolate the variable x.
    4. Solve for x.