Adding a Constant to Both Sides

Oct 13, 2025 by leolavanderia.com.br 31 views

Solving Quadratic Equations: A Step-by-Step Analysis

Solving quadratic equations is a fundamental concept in mathematics, and it's essential to understand the different methods and techniques involved. In this article, we will analyze the steps taken by five different students to solve the equation $$x^2 - 24x = 9$$. Our goal is to identify the correct steps and provide a clear explanation of the solution.

The given equation is $$x^2 - 24x = 9$$. This is a quadratic equation in the form of $$ax^2 + bx + c = 0$$, where $$a = 1$$, $$b = -24$$, and $$c = -9$$.

Hannah's step is as follows:

x^2 - 24x + 144 = 9

To analyze this step, let's examine the equation. Hannah has added $$144$$ to both sides of the equation, which is equivalent to adding $$144$$ to the left-hand side of the equation. This is a valid operation, but it's essential to understand the reasoning behind it.

Adding a Constant to Both Sides

When we add a constant to both sides of an equation, we are essentially adding the same value to both sides. In this case, Hannah has added $$144$$ to both sides, which is equivalent to adding $$144$$ to the left-hand side of the equation. This is a valid operation, but it's essential to understand the reasoning behind it.

The Correct Step

However, Hannah's step is not entirely correct. The correct step would be to add $$144$$ to both sides of the equation, but only if we are trying to complete the square. In this case, we are not trying to complete the square, so Hannah's step is not necessary.

Aaron's step is as follows:

x^2 - 24x + 144 = 9

Aaron's step is identical to Hannah's step, and for the same reasons, it's not entirely correct.

Michael's step is as follows:

(x-12)^2 - 144 = 9

Michael has taken a different approach by trying to complete the square. He has added $$144$$ to both sides of the equation, which is equivalent to adding $$144$$ to the left-hand side of the equation.

Completing the Square

Completing the square is a technique used to solve quadratic equations by rewriting them in a specific form. In this case, Michael has tried to complete the square by adding $$144$$ to both sides of the equation. This is a valid operation, but it's essential to understand the reasoning behind it.

The Correct Step

Michael's step is correct, but only if we are trying to complete the square. In this case, we are not trying to complete the square, so Michael's step is not necessary.

David's step is as follows:

(x-12)^2 - 144 = 9

David's step is identical to Michael's step, and for the same reasons, it's not entirely correct.

Emily's step is as follows:

(x-12)^2 - 144 = 9

Emily's step is identical to Michael's step, and for the same reasons, it's not entirely correct.

In conclusion, only two students have taken the correct step to solve the equation $$x^2 - 24x = 9$$. The correct steps are:

Completing the Square: This involves adding $$144$$ to both sides of the equation to rewrite it in a specific form.
Adding a Constant to Both Sides: This involves adding a constant to both sides of the equation, but only if we are trying to complete the square.

The other three students have taken incorrect steps, which are not necessary to solve the equation.

The final answer is not provided in this article, as the goal is to analyze the steps taken by the students rather than providing a final answer.

Based on the analysis, the following recommendations can be made:

Practice Completing the Square: Completing the square is a technique used to solve quadratic equations by rewriting them in a specific form. It's essential to practice this technique to become proficient in solving quadratic equations.
Understand the Reasoning Behind Each Step: It's essential to understand the reasoning behind each step taken to solve a quadratic equation. This will help you identify the correct steps and avoid making mistakes.

By following these recommendations, you can improve your skills in solving quadratic equations and become proficient in mathematics.
Quadratic Equations: A Q&A Guide

Quadratic equations are a fundamental concept in mathematics, and they can be challenging to solve. In our previous article, we analyzed the steps taken by five different students to solve the equation $$x^2 - 24x = 9$$. In this article, we will provide a Q&A guide to help you understand quadratic equations and solve them with confidence.

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 (usually x) is two. The general form of a quadratic equation is $$ax^2 + bx + c = 0$$, where $$a$$, $$b$$, and $$c$$ are constants.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods to solve a quadratic equation, including:

Factoring: This involves expressing the quadratic equation as a product of two binomials.
Completing the Square: This involves rewriting the quadratic equation in a specific form to solve it.
Using the Quadratic Formula: This involves using a formula to find the solutions of the quadratic equation.

Q: What is the Quadratic Formula?

A: The quadratic formula is a formula used to find the solutions of a quadratic equation. It is given by:

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

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to:

Identify the values of $$a$$, $$b$$, and $$c$$ in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression to find the solutions.

Q: What is Completing the Square?

A: Completing the square is a technique used to solve quadratic equations by rewriting them in a specific form. It involves adding a constant to both sides of the equation to create a perfect square trinomial.

Q: How Do I Complete the Square?

A: To complete the square, you need to:

Identify the values of $$a$$, $$b$$, and $$c$$ in the quadratic equation.
Add $$\left(\frac{b}{2}\right)^2$$ to both sides of the equation.
Simplify the expression to find the solutions.

Q: What is Factoring?

A: Factoring is a technique used to solve quadratic equations by expressing them as a product of two binomials.

Q: How Do I Factor a Quadratic Equation?

A: To factor a quadratic equation, you need to:

Identify the values of $$a$$, $$b$$, and $$c$$ in the quadratic equation.
Look for two binomials that multiply to give the quadratic expression.
Simplify the expression to find the solutions.

Q: What are the Common Mistakes to Avoid When Solving Quadratic Equations?

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

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying the expression.
Not checking the solutions: Make sure to check the solutions to ensure they are valid.
Not using the correct method: Make sure to use the correct method to solve the quadratic equation.

In conclusion, solving quadratic equations can be challenging, but with the right techniques and practice, you can become proficient in solving them. Remember to follow the order of operations, check the solutions, and use the correct method to solve the quadratic equation. By following these tips, you can improve your skills in solving quadratic equations and become confident in mathematics.

Practice, Practice, Practice: The more you practice solving quadratic equations, the more confident you will become.
Understand the Concepts: Make sure to understand the concepts behind solving quadratic equations, including factoring, completing the square, and using the quadratic formula.
Seek Help When Needed: Don't be afraid to seek help when needed. Ask your teacher, tutor, or classmate for assistance.

By following these tips, you can improve your skills in solving quadratic equations and become proficient in mathematics.