Divide the equation by the coefficient of the squared term

Oct 13, 2025 by windailys.com 10 views

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 explore the process of solving quadratic equations, with a focus on completing the square. We will use the equation $8x^2 - 48x = -104$ as a case study to demonstrate the steps involved in rewriting the equation in the form $(x \pm 3 \sqrt{ })^2 = -4$ and completing the square.

Rewriting the Equation

The given equation is $8x^2 - 48x = -104$ . To rewrite this equation in the form $(x \pm 3 \sqrt{ })^2 = -4$ , we need to follow these steps:

Divide the equation by the coefficient of the squared term: In this case, the coefficient of the squared term is 8. We will divide the entire equation by 8 to simplify it.

x^2 - 6x = -13

Identify the value of $a$ : In the equation $x^2 \pm 6x$ , the value of $a$ is 1.
Calculate the value of $b$ : In the equation $x^2 \pm 6x$ , the value of $b$ is 6.
Calculate the value of $c$ : In the equation $x^2 \pm 6x$ , the value of $c$ is 0.
Rewrite the equation in the form $(x \pm a)^2 = b^2 - c$ : Using the values of $a$ , $b$ , and $c$ calculated above, we can rewrite the equation as:

(x - 3)^2 = 9 - 13

Simplify the equation: Simplifying the equation, we get:

(x - 3)^2 = -4

Completing the Square

Now that we have rewritten the equation in the form $(x \pm a)^2 = b^2 - c$ , we can complete the square. To do this, we need to follow these steps:

Add and subtract the square of half the coefficient of the linear term: In this case, the coefficient of the linear term is -6. Half of this value is -3, and the square of -3 is 9. We will add and subtract 9 to the equation.

(x - 3)^2 - 9 = -4

Add 9 to both sides of the equation: To isolate the squared term, we will add 9 to both sides of the equation.

(x - 3)^2 = -4 + 9

Simplify the equation: Simplifying the equation, we get:

(x - 3)^2 = 5

Conclusion

In this article, we have demonstrated the process of rewriting a quadratic equation in the form $(x \pm a)^2 = b^2 - c$ and completing the square. We used the equation $8x^2 - 48x = -104$ as a case study to illustrate the steps involved in rewriting the equation and completing the square. By following these steps, we can solve quadratic equations and gain a deeper understanding of the underlying mathematics.

Discussion

What are some common mistakes to avoid when rewriting a quadratic equation in the form $(x \pm a)^2 = b^2 - c$ ?
How does completing the square relate to other methods of solving quadratic equations, such as factoring and the quadratic formula?
Can you think of any real-world applications of quadratic equations and completing the square?

Additional Resources

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

Final Thoughts

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 explored the process of rewriting a quadratic equation in the form $(x \pm a)^2 = b^2 - c$ and completing the square. In this article, we will answer some of the most frequently asked questions about quadratic equations and completing the square.

Q&A

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. It is typically written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I know if a quadratic equation can be factored?

A: A quadratic equation can be factored if it can be written in the form $(x - r)(x - s) = 0$ , where $r$ and $s$ are the roots of the equation. To determine if a quadratic equation can be factored, try to find two numbers whose product is $ac$ and whose sum is $b$ .

Q: What is the quadratic formula?

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

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

Q: How do I complete the square?

A: To complete the square, follow these steps:

Divide the equation by the coefficient of the squared term.
Identify the value of $a$ .
Calculate the value of $b$ .
Calculate the value of $c$ .
Rewrite the equation in the form $(x \pm a)^2 = b^2 - c$ .
Add and subtract the square of half the coefficient of the linear term.
Add the square of half the coefficient of the linear term to both sides of the equation.

Q: What are some common mistakes to avoid when rewriting a quadratic equation in the form $(x \pm a)^2 = b^2 - c$ ?

A: Some common mistakes to avoid when rewriting a quadratic equation in the form $(x \pm a)^2 = b^2 - c$ include:

Not dividing the equation by the coefficient of the squared term.
Not identifying the value of $a$ .
Not calculating the value of $b$ .
Not calculating the value of $c$ .
Not rewriting the equation in the correct form.

Q: How does completing the square relate to other methods of solving quadratic equations?

A: Completing the square is one of several methods of solving quadratic equations, including factoring and the quadratic formula. Each method has its own strengths and weaknesses, and the choice of method will depend on the specific equation and the desired solution.

Q: Can you think of any real-world applications of quadratic equations and completing the square?

A: Yes, quadratic equations and completing the square have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity and other forces.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

Quadratic equations and completing the square are fundamental concepts in mathematics, and solving them is a crucial skill for students and professionals alike. By following the steps outlined in this article, we can rewrite a quadratic equation in the form $(x \pm a)^2 = b^2 - c$ and complete the square. With practice and patience, we can master the art of solving quadratic equations and gain a deeper understanding of the underlying mathematics.

Discussion

What are some other methods of solving quadratic equations?
How does completing the square relate to other methods of solving quadratic equations?
Can you think of any real-world applications of quadratic equations and completing the square?

Additional Resources

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