A Student Is Deriving The Quadratic Formula. Her First Two Steps Are Shown:Step 1: { -c = Ax^2 + Bx$}$ 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.

Mar 13, 2025 by ADMIN 233 views

**Deriving the Quadratic Formula: Understanding the First Two Steps**

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 a powerful tool that helps us find the solutions to quadratic equations, and it is derived from the process of completing the square. In this article, we will explore the first two steps of deriving the quadratic formula, and we will examine the reasoning behind each step.

Step 1: The Original Equation

The first step in deriving the quadratic formula is to start with the original equation: ax^2 + bx + c = 0. This equation represents a quadratic equation in the form of a polynomial of degree two. The coefficients a, b, and c are constants, and x is the variable.

Step 2: Factoring Out the Coefficient a

The second step in deriving the quadratic formula is to factor out the coefficient a from the equation. This is done by multiplying the entire equation by 1/a, which is equivalent to dividing each term by a. The resulting equation is:

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

Justifying Step 2: The Division Property of Equality

The division property of equality states that if we divide both sides of an equation by a non-zero constant, the resulting equation is still true. In this case, we are dividing both sides of the equation by a, which is a non-zero constant. Therefore, we can justify Step 2 by applying the division property of equality.

Alternative Justification: Distributive Property

Another way to justify Step 2 is to use the distributive property of multiplication over addition. When we multiply a binomial by a constant, we can distribute the constant to each term in the binomial. In this case, we are multiplying the binomial (x^2 + b/a x) by a, which is a constant. Therefore, we can justify Step 2 by applying the distributive property of multiplication over addition.

Conclusion

In conclusion, the second step in deriving the quadratic formula is to factor out the coefficient a from the equation. This is done by multiplying the entire equation by 1/a, which is equivalent to dividing each term by a. We can justify this step by applying the division property of equality or the distributive property of multiplication over addition.

Deriving the Quadratic Formula: Completing the Square

Now that we have factored out the coefficient a, we can proceed to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial. The resulting equation is:

a\left(x + \frac{b}{2a}\right)^2 = c - \frac{b^2}{4a}

Solving for x

To solve for x, we can divide both sides of the equation by a, which gives us:

x + \frac{b}{2a} = \pm \sqrt{\frac{c - \frac{b^2}{4a}}{a}}

Simplifying the Expression

We can simplify the expression by multiplying both sides of the equation by 2a, which gives us:

2ax + b = \pm 2a\sqrt{\frac{c - \frac{b^2}{4a}}{a}}

Final Expression

The final expression for the quadratic formula is:

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

Conclusion

In conclusion, the quadratic formula is a powerful tool that helps us solve quadratic equations. It is derived from the process of completing the square, and it involves factoring out the coefficient a, adding and subtracting a constant term, and solving for x. We have examined the first two steps of deriving the quadratic formula, and we have justified each step using the division property of equality and the distributive property of multiplication over addition.

Applications of the Quadratic Formula

The quadratic formula has many applications in mathematics and science. It is used to solve quadratic equations, which are common in physics, engineering, and computer science. It is also used to model real-world problems, such as the motion of objects under the influence of gravity or the growth of populations.

Real-World Examples

Here are a few real-world examples of the quadratic formula in action:

Projectile Motion: The quadratic formula is used to model the motion of projectiles under the influence of gravity. For example, the trajectory of a thrown ball can be modeled using the quadratic formula.
Population Growth: The quadratic formula is used to model the growth of populations. For example, the population of a city can be modeled using the quadratic formula.
Electrical Circuits: The quadratic formula is used to model the behavior of electrical circuits. For example, the voltage and current in a circuit can be modeled using the quadratic formula.

Conclusion

In conclusion, the quadratic formula is a powerful tool that has many applications in mathematics and science. It is used to solve quadratic equations, model real-world problems, and make predictions about the behavior of complex systems. We have examined the first two steps of deriving the quadratic formula, and we have justified each step using the division property of equality and the distributive property of multiplication over addition.

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 a powerful tool that helps us find the solutions to quadratic equations, and it is derived from the process of completing the square. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations of the form ax^2 + bx + c = 0. It is a powerful tool that helps us find the solutions to quadratic equations.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula, and then simplify the expression. The formula is:

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

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

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves finding the factors of the quadratic expression, while the quadratic formula involves using a formula to find the solutions.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic expression cannot be factored easily, or when you need to find the solutions to a quadratic equation quickly.

Q: Can I use the quadratic formula to solve quadratic inequalities?

A: No, the quadratic formula is only used to solve quadratic equations, not quadratic inequalities. To solve quadratic inequalities, you need to use a different method, such as graphing or using a calculator.

Q: How do I know if the quadratic formula will give me real solutions?

A: To determine if the quadratic formula will give you real solutions, you need to check the discriminant, which is the expression under the square root. If the discriminant is positive, then the quadratic formula will give you two real solutions. If the discriminant is zero, then the quadratic formula will give you one real solution. If the discriminant is negative, then the quadratic formula will give you two complex solutions.

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

A: No, the quadratic formula is only used to solve quadratic equations, not cubic equations. To solve cubic equations, you need to use a different method, such as factoring or using a calculator.

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you need to plug in the values of a, b, and c into the formula, and then simplify the expression. You can also use a calculator to simplify the expression.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression, as complex numbers can be tricky to work with.

Conclusion

In conclusion, the quadratic formula is a powerful tool that helps us solve quadratic equations. It is a fundamental concept in algebra, and it has many applications in mathematics and science. We have answered some of the most frequently asked questions about the quadratic formula, and we hope that this article has been helpful to you.

Additional Resources

If you want to learn more about the quadratic formula, here are some additional resources that you may find helpful:

Textbooks: There are many textbooks that cover the quadratic formula, including "Algebra" by Michael Artin and "College Algebra" by James Stewart.
Online Resources: There are many online resources that cover the quadratic formula, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
Videos: There are many videos that cover the quadratic formula, including Crash Course, 3Blue1Brown, and Vi Hart.
Practice Problems: There are many practice problems that you can use to practice using the quadratic formula, including those found on Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.