What Is The Intermediate Step In The Form \[$(x+a)^2=b\$\] As A Result Of Completing The Square For The Following Equation?$\[4x^2 - 72x = -68\\]

Feb 28, 2025 by ADMIN 146 views

**What is the Intermediate Step in Completing the Square for the Given Equation?**

Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in the form of a perfect square trinomial. In this article, we will explore the intermediate step in completing the square for the given equation $4x^2 - 72x = -68$ . We will break down the process step by step and provide a clear understanding of the intermediate step involved.

Understanding the Given Equation

The given equation is $4x^2 - 72x = -68$ . To complete the square, we need to manipulate this equation to express it in the form of a perfect square trinomial. The first step is to isolate the quadratic term on one side of the equation.

Step 1: Isolate the Quadratic Term

We can start by adding $68$ to both sides of the equation to isolate the quadratic term.

4x^2 - 72x + 68 = 0

Step 2: Factor Out the Coefficient of the Quadratic Term

Next, we need to factor out the coefficient of the quadratic term, which is $4$ . This will help us to create a perfect square trinomial.

4(x^2 - 18x) + 68 = 0

Step 3: Add and Subtract the Square of Half the Coefficient of the Linear Term

To complete the square, we need to add and subtract the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is $-18$ , and half of it is $-9$ . The square of $-9$ is $81$ .

4(x^2 - 18x + 81 - 81) + 68 = 0

Step 4: Simplify the Expression

Now, we can simplify the expression by combining like terms.

4(x^2 - 18x + 81) - 324 + 68 = 0

4(x^2 - 18x + 81) - 256 = 0

Step 5: Factor the Perfect Square Trinomial

The expression $x^2 - 18x + 81$ is a perfect square trinomial, and we can factor it as $(x - 9)^2$ .

4(x - 9)^2 - 256 = 0

Step 6: Add 256 to Both Sides of the Equation

Finally, we can add $256$ to both sides of the equation to get the final result.

4(x - 9)^2 = 256

Step 7: Divide Both Sides of the Equation by 4

To isolate the perfect square trinomial, we can divide both sides of the equation by $4$ .

(x - 9)^2 = 64

Step 8: Take the Square Root of Both Sides of the Equation

Taking the square root of both sides of the equation, we get:

x - 9 = \pm \sqrt{64}

x - 9 = \pm 8

Step 9: Add 9 to Both Sides of the Equation

Finally, we can add $9$ to both sides of the equation to get the final result.

x = 9 \pm 8

x = 17, x = 1

Conclusion

In this article, we explored the intermediate step in completing the square for the given equation $4x^2 - 72x = -68$ . We broke down the process step by step and provided a clear understanding of the intermediate step involved. The final result is $x = 17, x = 1$ , which are the solutions to the given equation.

Key Takeaways

Completing the square is a powerful technique used to solve quadratic equations.
The intermediate step in completing the square involves adding and subtracting the square of half the coefficient of the linear term.
The perfect square trinomial can be factored as $(x - a)^2$ , where $a$ is the value that is being subtracted from $x$ .
The final result is obtained by adding the square of half the coefficient of the linear term to both sides of the equation.

Frequently Asked Questions

What is completing the square?
- Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in the form of a perfect square trinomial.
What is the intermediate step in completing the square?
- The intermediate step in completing the square involves adding and subtracting the square of half the coefficient of the linear term.
How do I factor the perfect square trinomial?
- The perfect square trinomial can be factored as $(x - a)^2$ , where $a$ is the value that is being subtracted from $x$ .

References

Completing the Square
Perfect Square Trinomial
Quadratic Equations
Frequently Asked Questions: Completing the Square =====================================================

Introduction

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in the form of a perfect square trinomial.

Q: What is the intermediate step in completing the square?

A: The intermediate step in completing the square involves adding and subtracting the square of half the coefficient of the linear term.

Q: How do I factor the perfect square trinomial?

A: The perfect square trinomial can be factored as $(x - a)^2$ , where $a$ is the value that is being subtracted from $x$ .

Q: What is the difference between completing the square and factoring?

A: Completing the square involves manipulating the equation to express it in the form of a perfect square trinomial, while factoring involves expressing the quadratic expression as a product of two binomials.

Q: Can I use completing the square to solve all quadratic equations?

A: No, completing the square is not suitable for all quadratic equations. It is particularly useful for equations that can be expressed in the form of $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I determine if an equation can be solved using completing the square?

A: To determine if an equation can be solved using completing the square, you need to check if the equation can be expressed in the form of $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: What are the advantages of completing the square?

A: The advantages of completing the square include:

It is a powerful technique for solving quadratic equations.
It can be used to find the solutions of quadratic equations that are not easily solvable using other methods.
It can be used to find the solutions of quadratic equations that have complex roots.

Q: What are the disadvantages of completing the square?

A: The disadvantages of completing the square include:

It can be a time-consuming process.
It requires a good understanding of algebraic manipulations.
It may not be suitable for all types of quadratic equations.

Q: Can I use completing the square to solve quadratic equations with complex roots?

A: Yes, completing the square can be used to solve quadratic equations with complex roots.

Q: How do I use completing the square to solve quadratic equations with complex roots?

A: To use completing the square to solve quadratic equations with complex roots, you need to follow the same steps as for solving quadratic equations with real roots.

Q: What are some common mistakes to avoid when using completing the square?

A: Some common mistakes to avoid when using completing the square include:

Not checking if the equation can be expressed in the form of $ax^2 + bx + c = 0$ .
Not adding and subtracting the square of half the coefficient of the linear term correctly.
Not factoring the perfect square trinomial correctly.

Conclusion

In this article, we have answered some of the most frequently asked questions about completing the square. We have discussed the advantages and disadvantages of completing the square, and provided some tips on how to use it to solve quadratic equations with complex roots. We hope that this article has been helpful in providing a better understanding of completing the square.

Key Takeaways

Completing the square is a powerful technique for solving quadratic equations.
It involves manipulating the equation to express it in the form of a perfect square trinomial.
It can be used to find the solutions of quadratic equations that are not easily solvable using other methods.
It can be used to find the solutions of quadratic equations that have complex roots.

Frequently Asked Questions

What is completing the square?
- Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in the form of a perfect square trinomial.
What is the intermediate step in completing the square?
- The intermediate step in completing the square involves adding and subtracting the square of half the coefficient of the linear term.
How do I factor the perfect square trinomial?
- The perfect square trinomial can be factored as $(x - a)^2$ , where $a$ is the value that is being subtracted from $x$ .