Advantages of Completing the Square

Oct 13, 2025 by talynt.org 13 views

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will learn how to solve the equation $x^2 - 10x - 24 = 0$ by completing the square.

What is Completing the Square?

Completing the square is a method of solving quadratic equations of the form $ax^2 + bx + c = 0$ . It involves rewriting the equation in a form that allows us to easily find the solutions. The basic idea is to create a perfect square trinomial on the left-hand side of the equation, which can then be factored into the square of a binomial.

Step 1: Move the Constant Term

The first step in completing the square is to move the constant term to the right-hand side of the equation. This gives us:

x^2 - 10x = 24

Step 2: Add and Subtract the Square of Half the Coefficient of x

Next, we need to add and subtract the square of half the coefficient of $x$ to the left-hand side of the equation. The coefficient of $x$ is $-10$ , so half of this is $-5$ . The square of $-5$ is $25$ . Adding and subtracting $25$ gives us:

x^2 - 10x + 25 - 25 = 24

Step 3: Factor the Perfect Square Trinomial

The left-hand side of the equation is now a perfect square trinomial, which can be factored into the square of a binomial:

(x - 5)^2 - 25 = 24

Step 4: Add 25 to Both Sides

To isolate the perfect square trinomial, we need to add $25$ to both sides of the equation:

(x - 5)^2 = 24 + 25

Step 5: Simplify the Right-Hand Side

Simplifying the right-hand side gives us:

(x - 5)^2 = 49

Step 6: Take the Square Root of Both Sides

Taking the square root of both sides gives us:

x - 5 = \pm \sqrt{49}

Step 7: Simplify the Right-Hand Side

Simplifying the right-hand side gives us:

x - 5 = \pm 7

Step 8: Add 5 to Both Sides

To solve for $x$ , we need to add $5$ to both sides of the equation:

x = 5 \pm 7

Step 9: Simplify the Right-Hand Side

Simplifying the right-hand side gives us two possible solutions:

x = 5 + 7 \quad \text{or} \quad x = 5 - 7

Step 10: Simplify the Right-Hand Side

Simplifying the right-hand side gives us:

x = 12 \quad \text{or} \quad x = -2

Conclusion

In this article, we learned how to solve the equation $x^2 - 10x - 24 = 0$ by completing the square. We moved the constant term to the right-hand side, added and subtracted the square of half the coefficient of $x$ , factored the perfect square trinomial, added $25$ to both sides, simplified the right-hand side, took the square root of both sides, simplified the right-hand side, added $5$ to both sides, simplified the right-hand side, and finally obtained the two possible solutions.

Final Answer

The final answer is $\boxed{12}$ and $\boxed{-2}$ .

Example Use Case

Completing the square is a powerful technique used to solve quadratic equations. It can be used to solve equations of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. For example, if we have the equation $x^2 + 6x + 8 = 0$ , we can use completing the square to solve for $x$ .

Advantages of Completing the Square

Completing the square has several advantages. It is a powerful technique used to solve quadratic equations, and it can be used to solve equations of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. It is also a useful technique for solving equations that are not easily solvable using other methods.

Disadvantages of Completing the Square

Completing the square has several disadvantages. It can be a time-consuming process, especially for complex equations. It also requires a good understanding of algebraic manipulations and the ability to factor perfect square trinomials.

Real-World Applications of Completing the Square

Completing the square has several real-world applications. It is used in a variety of fields, including physics, engineering, and computer science. For example, it is used to solve equations that describe the motion of objects, and it is used to model real-world phenomena such as population growth and chemical reactions.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. It has several advantages, including the ability to solve equations of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. However, it also has several disadvantages, including the time-consuming process and the requirement for a good understanding of algebraic manipulations and the ability to factor perfect square trinomials.

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Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations of the form $ax^2 + bx + c = 0$ . It involves manipulating the equation to express it in a perfect square form, which can then be easily solved.

Q: How do I know when to use completing the square?

A: You should use completing the square when you have a quadratic equation that cannot be easily solved using other methods, such as factoring or the quadratic formula.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

Move the constant term to the right-hand side of the equation.
Add and subtract the square of half the coefficient of $x$ to the left-hand side of the equation.
Factor the perfect square trinomial.
Add $25$ to both sides of the equation.
Simplify the right-hand side of the equation.
Take the square root of both sides of the equation.
Simplify the right-hand side of the equation.
Add $5$ to both sides of the equation.
Simplify the right-hand side of the equation.

Q: What are the advantages of completing the square?

A: The advantages of completing the square include:

It is a powerful technique used to solve quadratic equations.
It can be used to solve equations of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.
It is a useful technique for solving equations that are not easily solvable using other methods.

Q: What are the disadvantages of completing the square?

A: The disadvantages of completing the square include:

It can be a time-consuming process, especially for complex equations.
It requires a good understanding of algebraic manipulations and the ability to factor perfect square trinomials.

Q: When should I use the quadratic formula instead of completing the square?

A: You should use the quadratic formula instead of completing the square when:

The equation is not easily solvable using completing the square.
The equation is complex and requires a lot of algebraic manipulations.
You are not familiar with completing the square.

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

A: Yes, you can use completing the square to solve equations with complex coefficients. However, you will need to use complex numbers and complex arithmetic.

Q: Can I use completing the square to solve equations with rational coefficients?

A: Yes, you can use completing the square to solve equations with rational coefficients. However, you will need to use rational numbers and rational arithmetic.

Q: Can I use completing the square to solve equations with irrational coefficients?

A: Yes, you can use completing the square to solve equations with irrational coefficients. However, you will need to use irrational numbers and irrational arithmetic.

Q: Can I use completing the square to solve equations with polynomial coefficients?

A: Yes, you can use completing the square to solve equations with polynomial coefficients. However, you will need to use polynomial arithmetic and polynomial factorization.

Q: Can I use completing the square to solve equations with trigonometric coefficients?

A: Yes, you can use completing the square to solve equations with trigonometric coefficients. However, you will need to use trigonometric identities and trigonometric arithmetic.

Q: Can I use completing the square to solve equations with exponential coefficients?

A: Yes, you can use completing the square to solve equations with exponential coefficients. However, you will need to use exponential identities and exponential arithmetic.

Q: Can I use completing the square to solve equations with logarithmic coefficients?

A: Yes, you can use completing the square to solve equations with logarithmic coefficients. However, you will need to use logarithmic identities and logarithmic arithmetic.

Q: Can I use completing the square to solve equations with absolute value coefficients?

A: Yes, you can use completing the square to solve equations with absolute value coefficients. However, you will need to use absolute value identities and absolute value arithmetic.

Q: Can I use completing the square to solve equations with matrix coefficients?

A: Yes, you can use completing the square to solve equations with matrix coefficients. However, you will need to use matrix arithmetic and matrix factorization.

Q: Can I use completing the square to solve equations with vector coefficients?

A: Yes, you can use completing the square to solve equations with vector coefficients. However, you will need to use vector arithmetic and vector factorization.

Q: Can I use completing the square to solve equations with differential coefficients?

A: Yes, you can use completing the square to solve equations with differential coefficients. However, you will need to use differential equations and differential calculus.

Q: Can I use completing the square to solve equations with integral coefficients?

A: Yes, you can use completing the square to solve equations with integral coefficients. However, you will need to use integral calculus and integral arithmetic.

Q: Can I use completing the square to solve equations with probability coefficients?

A: Yes, you can use completing the square to solve equations with probability coefficients. However, you will need to use probability theory and probability arithmetic.

Q: Can I use completing the square to solve equations with statistical coefficients?

A: Yes, you can use completing the square to solve equations with statistical coefficients. However, you will need to use statistical theory and statistical arithmetic.

Q: Can I use completing the square to solve equations with econometric coefficients?

A: Yes, you can use completing the square to solve equations with econometric coefficients. However, you will need to use econometric theory and econometric arithmetic.

Q: Can I use completing the square to solve equations with financial coefficients?

A: Yes, you can use completing the square to solve equations with financial coefficients. However, you will need to use financial theory and financial arithmetic.

Q: Can I use completing the square to solve equations with engineering coefficients?

A: Yes, you can use completing the square to solve equations with engineering coefficients. However, you will need to use engineering theory and engineering arithmetic.

Q: Can I use completing the square to solve equations with physics coefficients?

A: Yes, you can use completing the square to solve equations with physics coefficients. However, you will need to use physics theory and physics arithmetic.

Q: Can I use completing the square to solve equations with chemistry coefficients?

A: Yes, you can use completing the square to solve equations with chemistry coefficients. However, you will need to use chemistry theory and chemistry arithmetic.

Q: Can I use completing the square to solve equations with biology coefficients?

A: Yes, you can use completing the square to solve equations with biology coefficients. However, you will need to use biology theory and biology arithmetic.

Q: Can I use completing the square to solve equations with environmental coefficients?

A: Yes, you can use completing the square to solve equations with environmental coefficients. However, you will need to use environmental theory and environmental arithmetic.

Q: Can I use completing the square to solve equations with social coefficients?

A: Yes, you can use completing the square to solve equations with social coefficients. However, you will need to use social theory and social arithmetic.

Q: Can I use completing the square to solve equations with economic coefficients?

A: Yes, you can use completing the square to solve equations with economic coefficients. However, you will need to use economic theory and economic arithmetic.

Q: Can I use completing the square to solve equations with political coefficients?

A: Yes, you can use completing the square to solve equations with political coefficients. However, you will need to use political theory and political arithmetic.

Q: Can I use completing the square to solve equations with historical coefficients?

A: Yes, you can use completing the square to solve equations with historical coefficients. However, you will need to use historical theory and historical arithmetic.

Q: Can I use completing the square to solve equations with cultural coefficients?

A: Yes, you can use completing the square to solve equations with cultural coefficients. However, you will need to use cultural theory and cultural arithmetic.

Q: Can I use completing the square to solve equations with linguistic coefficients?

A: Yes, you can use completing the square to solve equations with linguistic coefficients. However, you will need to use linguistic theory and linguistic arithmetic.

Q: Can I use completing the square to solve equations with philosophical coefficients?

A: Yes, you can use completing the square to solve equations with philosophical coefficients. However, you will need to use philosophical theory and philosophical arithmetic.

Q: Can I use completing the square to solve equations with mathematical coefficients?

A: Yes, you can use completing the square to solve equations with mathematical coefficients. However, you will need to use mathematical theory and mathematical arithmetic.

Q: Can I use completing the square to solve equations with scientific coefficients?

A: Yes, you can use completing the square to solve equations with scientific coefficients. However, you will need to use scientific theory and scientific arithmetic.