What Is The Intermediate Step In The Form $(x+a)^2=b$ As A Result Of Completing The Square For The Following Equation?$x^2 + 37 = -14x - 8$

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

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 be easily solved. In this article, we will explore the intermediate step in completing the square for the given equation x2+37=βˆ’14xβˆ’8x^2 + 37 = -14x - 8.

The given equation is x2+37=βˆ’14xβˆ’8x^2 + 37 = -14x - 8. To complete the square, we need to move the constant term to the right-hand side of the equation. This gives us x2+14x+37=βˆ’8x^2 + 14x + 37 = -8.

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 x2+14x+37=βˆ’8x^2 + 14x + 37 = -8.

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

The next step is to add and subtract the square of half the coefficient of xx to the left-hand side of the equation. The coefficient of xx is 14, so half of it is 7. The square of 7 is 49. Adding and subtracting 49 to the left-hand side of the equation gives us:

x2+14x+49βˆ’49+37=βˆ’8x^2 + 14x + 49 - 49 + 37 = -8

Step 3: Simplify the Equation

Now, we can simplify the equation by combining like terms. The left-hand side of the equation becomes:

(x+7)2βˆ’12=βˆ’8(x + 7)^2 - 12 = -8

Step 4: Add 12 to Both Sides

The next step is to add 12 to both sides of the equation to isolate the squared term. This gives us:

(x+7)2=βˆ’8+12(x + 7)^2 = -8 + 12

Step 5: Simplify the Right-Hand Side

Simplifying the right-hand side of the equation gives us:

(x+7)2=4(x + 7)^2 = 4

The Intermediate Step

The intermediate step in completing the square for the given equation is:

(x+7)2=4(x + 7)^2 = 4

This is the result of adding and subtracting the square of half the coefficient of xx to the left-hand side of the equation.

In conclusion, the intermediate step in completing the square for the given equation x2+37=βˆ’14xβˆ’8x^2 + 37 = -14x - 8 is (x+7)2=4(x + 7)^2 = 4. This step involves adding and subtracting the square of half the coefficient of xx to the left-hand side of the equation, which allows us to express the equation in a perfect square form.

The final answer is: (x+7)2=4\boxed{(x + 7)^2 = 4}
Completing the Square: A Q&A Guide

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 be easily solved. In this article, we will explore the intermediate step in completing the square for the given equation x2+37=βˆ’14xβˆ’8x^2 + 37 = -14x - 8. We will also answer some frequently asked questions about completing the square.

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 a perfect square form. This involves adding and subtracting a constant term to the left-hand side of the equation, which allows us to express the equation in a perfect square form.

Q: Why do we need to complete the square?

A: We need to complete the square because it allows us to express the equation in a perfect square form, which can be easily solved. This is particularly useful when solving quadratic equations that cannot be factored easily.

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

A: The intermediate step in completing the square is to add and subtract the square of half the coefficient of xx to the left-hand side of the equation. This allows us to express the equation in a perfect square form.

Q: How do I know when to add and subtract the square of half the coefficient of xx?

A: You know when to add and subtract the square of half the coefficient of xx when you have a quadratic equation in the form x2+bx+c=dx^2 + bx + c = d. In this case, you need to add and subtract the square of half the coefficient of xx to the left-hand side of the equation.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

(x+b2)2=c+b24(x + \frac{b}{2})^2 = c + \frac{b^2}{4}

Q: How do I apply the formula for completing the square?

A: To apply the formula for completing the square, you need to follow these steps:

  1. Move the constant term to the right-hand side of the equation.
  2. Add and subtract the square of half the coefficient of xx to the left-hand side of the equation.
  3. Simplify the equation by combining like terms.
  4. Add the square of half the coefficient of xx to both sides of the equation.
  5. Simplify the right-hand side of the equation.

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

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

  • Not moving the constant term to the right-hand side of the equation.
  • Not adding and subtracting the square of half the coefficient of xx to the left-hand side of the equation.
  • Not simplifying the equation by combining like terms.
  • Not adding the square of half the coefficient of xx to both sides of the equation.

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 be easily solved. By following the steps outlined in this article, you can complete the square and solve quadratic equations with ease.

The final answer is: (x+7)2=4\boxed{(x + 7)^2 = 4}