The Perfect Addition to Complete the Square: x² - 10x + __ = 7

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Completing the square is a powerful algebraic technique that can help us solve quadratic equations and unlock the mysteries hidden within them. In the case of the equation X2 – 10x = 7, we find ourselves on the cusp of discovering the missing link that will transform this equation into a perfect square trinomial. But what exactly is this missing number? What number should be added to both sides of the equation to complete the square? Brace yourself as we embark on a journey of mathematical exploration to unravel this enigma and witness the magic that lies within completing the square.


Introduction

In algebra, completing the square is a technique used to solve quadratic equations. It involves transforming an equation into a perfect square trinomial, which can then be easily solved. One key step in this process is determining the number that should be added to both sides of the equation to complete the square. In this article, we will explore how to complete the square for the equation x^2 - 10x = 7.

Understanding the Equation

To begin, let's take a closer look at the given equation: x^2 - 10x = 7. This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = -10, and c = 7. Our goal is to transform it into a perfect square trinomial by adding a certain number to both sides.

Determining the Coefficient of the Middle Term

Before we can determine the number to add, we must first find the coefficient of the middle term (bx). In our equation, b = -10. To find the coefficient, we divide b by 2 and square the result. So, (-10/2)^2 = (-5)^2 = 25. Therefore, the coefficient of the middle term is 25.

Adding the Number to Both Sides

Now that we know the coefficient of the middle term, we can add its square to both sides of the equation. Adding 25 to both sides of x^2 - 10x = 7 gives us x^2 - 10x + 25 = 32.

Completing the Square

The left side of the equation, x^2 - 10x + 25, can now be simplified as a perfect square trinomial. It is equal to (x - 5)^2. Therefore, our equation becomes (x - 5)^2 = 32.

Isolating the Square Term

To solve for x, we need to isolate the square term on one side of the equation. In this case, we already have (x - 5)^2 isolated on the left side, so our equation is already in the desired form.

Taking the Square Root

Next, we can take the square root of both sides to eliminate the exponent and solve for x. Taking the square root of (x - 5)^2 gives us x - 5, and the square root of 32 is approximately ±5.66.

Two Possible Solutions

Considering the positive square root, we have x - 5 = 5.66. Adding 5 to both sides, we get x = 10.66 as one possible solution.

For the negative square root, we have x - 5 = -5.66. Adding 5 to both sides, we get x = -0.66 as the other possible solution.

Verifying the Solutions

To ensure the accuracy of our solutions, we can substitute them back into the original equation. Plugging x = 10.66 into the equation x^2 - 10x = 7 results in 10.66^2 - 10(10.66) = 7, which is true. Similarly, substituting x = -0.66 gives us (-0.66)^2 - 10(-0.66) = 7, which is also true.

Conclusion

In conclusion, to complete the square for the equation x^2 - 10x = 7, we added 25 to both sides. This transformed the equation into a perfect square trinomial, (x - 5)^2 = 32. By taking the square root and solving for x, we found two possible solutions: x = 10.66 and x = -0.66. It is important to verify these solutions by substituting them back into the original equation.


Introduction

Completing the square is a fundamental concept in algebraic equations that involves transforming a quadratic equation into a perfect square trinomial. This process allows us to find the solutions to the equation and better understand its properties. In this discussion, we will explore the steps involved in completing the square for the equation x^2 – 10x = 7.

Understanding the given equation

The equation x^2 – 10x = 7 is a quadratic equation with a variable term, x^2, and a linear term, -10x. Our goal is to transform this equation into a perfect square trinomial, which will make it easier to solve and analyze.

Goal of completing the square

The purpose of completing the square is to determine the number that should be added to both sides of the equation in order to create a perfect square trinomial on the left side. This step simplifies the equation and allows us to easily factor or solve for the variable.

Identifying the coefficient of x

In the given equation x^2 – 10x = 7, the coefficient of x is -10. This value is crucial in completing the square as it helps us calculate the term that needs to be added to both sides of the equation.

Half of the coefficient squared

To find the number that should be added, we take half of the coefficient (-10/2) and square it. In this case, (-10/2)^2 equals 25.

Adding the perfect square trinomial to both sides

To complete the square, we add the perfect square trinomial, 25, to both sides of the equation. This step ensures that the left side of the equation becomes a perfect square trinomial.

Adding 25 to both sides of the equation x^2 – 10x = 7, we get:

x^2 – 10x + 25 = 7 + 25

Completing the square

By adding 25 to both sides of the equation, we have transformed the left side into a perfect square trinomial. The expression (x – 5)^2 represents the process of completing the square.

Simplifying the equation

After completing the square, we simplify the equation by combining like terms on the right side. In this case, 7 + 25 equals 32.

Therefore, our equation becomes:

(x – 5)^2 = 32

Solving for x

To find the value of x, we take the square root of both sides of the equation:

√((x – 5)^2) = √32

Taking the square root of the perfect square trinomial (x – 5)^2 gives us:

x – 5 = ±√32

Obtaining the final solution

To determine the two possible solutions for x, we consider both the positive and negative square root values from the equation (x – 5)^2 = 32.

For the positive square root:

x – 5 = √32

For the negative square root:

x – 5 = -√32Solving for x in each case, we add 5 to both sides of the equations:

For the positive square root:

x = 5 + √32

For the negative square root:

x = 5 - √32Therefore, the equation x^2 – 10x = 7 can be completed by adding 25 to both sides, resulting in (x – 5)^2 = 32. The solutions for x are x = 5 + √32 and x = 5 - √32.

What Number Should Be Added To Both Sides Of The Equation To Complete The Square?

Story:

Once upon a time, in a small village called Mathland, there lived a young mathematician named Alice. She was known for her exceptional problem-solving skills and her insatiable curiosity for all things mathematical.

One day, Alice came across an interesting equation written on the village noticeboard. It said, 'x^2 – 10x = 7'. Immediately, her analytical mind started working on solving this equation.

She knew that completing the square was a powerful technique to simplify quadratic equations. To do so, Alice needed to add a specific number to both sides of the equation.

As she pondered over the equation, Alice decided to break it down step by step to find the missing number:

Step 1:

Alice noticed that the coefficient of the 'x' term is -10, which is twice the product of the coefficients of x in the original equation. She divided -10 by 2, resulting in -5.

Step 2:

To complete the square, Alice squared the number obtained in Step 1, which gave her (-5)^2 = 25.

Step 3:

Now, Alice added the number obtained in Step 2 to both sides of the equation:

x^2 – 10x + 25 = 7 + 25

x^2 – 10x + 25 = 32

Step 4:

Alice realized that the left side of the equation could be rewritten as a perfect square: (x - 5)^2.

Step 5:

To solve the equation, Alice took the square root of both sides:

x - 5 = ±√32

Step 6:

Finally, Alice added 5 to both sides of the equation to isolate 'x' and find the solutions:

x = 5 ± √32

Alice felt a sense of accomplishment as she unraveled the mystery behind completing the square. She now understood how adding the missing number, 25 in this case, helped transform the equation into a perfect square trinomial.

With her newfound knowledge, Alice continued her mathematical journey, eager to face more challenges and unravel the secrets hidden within equations.

Table Information:

Below is a summary of the steps taken by Alice to complete the square for the equation x^2 – 10x = 7:

  1. Identify the coefficient of the 'x' term: -10
  2. Divide the coefficient by 2: -10 ÷ 2 = -5
  3. Square the result from step 2: (-5)^2 = 25
  4. Add the squared number to both sides of the equation: x^2 – 10x + 25 = 7 + 25
  5. Rewrite the left side of the equation as a perfect square: (x - 5)^2 = 32
  6. Take the square root of both sides of the equation: x - 5 = ±√32
  7. Add 5 to both sides of the equation: x = 5 ± √32

Thank you for visiting our blog! In this article, we discussed the concept of completing the square in algebraic equations. Specifically, we focused on the equation X^2 - 10x = 7 and explored how we can determine the number that needs to be added to both sides of the equation in order to complete the square.

To complete the square in this equation, we need to find a constant term that, when added to both sides, will allow us to factor the left-hand side of the equation into a perfect square trinomial. This process helps us solve quadratic equations and find their vertex form.

In this case, to complete the square in the equation X^2 - 10x = 7, we can follow a step-by-step approach. First, we need to divide the coefficient of the x-term by 2, which yields -5. Then, we square this value to obtain 25. Finally, we add 25 to both sides of the equation to complete the square. The equation now becomes X^2 - 10x + 25 = 7 + 25, which simplifies to (X - 5)^2 = 32.

Completing the square allows us to rewrite the equation in vertex form, which is a useful form for solving quadratic equations and graphing parabolas. In this case, the vertex form of the equation X^2 - 10x = 7 is (X - 5)^2 = 32. From here, we can easily identify the vertex of the parabola, which is located at the point (5, 32).

We hope this article has provided you with a clear understanding of how to complete the square in algebraic equations. By following the steps outlined above, you can confidently determine the number that needs to be added to both sides of the equation to complete the square. Feel free to explore our other articles for more insights into various mathematical concepts. Thank you once again for visiting our blog!


What Number Should Be Added To Both Sides Of The Equation To Complete The Square?

1. Explanation

Completing the square is a technique used in algebra to convert a quadratic equation into a perfect square trinomial. This process involves adding a specific number to both sides of the equation in order to create a perfect square trinomial on the left-hand side.

Example equation:

x^2 - 10x = 7

2. Steps to Complete the Square

To complete the square, follow these steps:

  1. Move the constant term (the number without a variable) to the opposite side of the equation. In our example, subtract 7 from both sides:
    • x^2 - 10x - 7 = 0
  2. Take half of the coefficient of the x-term (10/2 = 5) and square it (5^2 = 25).
  3. Add this squared value (25) to both sides of the equation:
    • x^2 - 10x + 25 - 7 = 25
    • x^2 - 10x + 18 = 25
  4. Factor the left-hand side of the equation into a perfect square trinomial:
    • (x - 5)^2 = 25
  5. Take the square root of both sides of the equation:
    • x - 5 = ±5
  6. Solve for x:
    • x = 5 ± 5
    • x = 10 or x = 0

3. Answer to What Number Should Be Added To Both Sides Of The Equation To Complete The Square?

In order to complete the square for the equation x^2 - 10x = 7, the number 18 should be added to both sides of the equation.