Let X 2 + 15 X = 49 X^2 + 15x = 49 X 2 + 15 X = 49 .What Values Make An Equivalent Number Sentence After Completing The Square?Enter Your Answers In The Boxes. X 2 + 15 X + □ = □ X^2 + 15x + \square = \square X 2 + 15 X + □ = □

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 be easily solved. In this article, we will explore the concept of completing the square and apply it to the given equation $x^2 + 15x = 49$. We will also discuss the values that make an equivalent number sentence after completing the square.

What is Completing the Square?

Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial. The constant term is called the "square root of the constant term" and is added and subtracted to both sides of the equation.

The Process of Completing the Square

The process of completing the square involves the following steps:

1. Write the equation in the form $ax^2 + bx + c = 0$: The given equation is already in this form, $x^2 + 15x = 49$.
2. Move the constant term to the right-hand side: Subtract 49 from both sides of the equation to get $x^2 + 15x - 49 = 0$.
3. Find the square root of the coefficient of the $x$ term: The coefficient of the $x$ term is 15, so we need to find the square root of 15. The square root of 15 is approximately 3.87.
4. Add and subtract the square of the square root to both sides: Add $(3.87)^2$ to both sides of the equation to get $x^2 + 15x + (3.87)^2 - (3.87)^2 - 49 = 0$.
5. Simplify the equation: Simplify the equation to get $x^2 + 15x + 15 - 49 = 0$.
6. Write the equation in the form $(x + a)^2 = b$: The equation can be written as $(x + 7.5)^2 = 34$.

Solving the Equation

Now that we have the equation in the form $(x + a)^2 = b$, we can solve for $x$. To do this, we need to take the square root of both sides of the equation.

The Square Root of Both Sides

Taking the square root of both sides of the equation gives us:

$$x + 7.5 = \pm \sqrt{34}$$

Solving for $x$

Now that we have the equation in the form $x + a = \pm \sqrt{b}$, we can solve for $x$. To do this, we need to isolate $x$ by subtracting $a$ from both sides of the equation.

The Solution

Subtracting 7.5 from both sides of the equation gives us:

$$x = -7.5 \pm \sqrt{34}$$

Conclusion

In this article, we have explored the concept of completing the square and applied it to the given equation $x^2 + 15x = 49$. We have also discussed the values that make an equivalent number sentence after completing the square. The process of completing the square involves rewriting the equation in a perfect square form by adding and subtracting a constant term. We have also solved the equation by taking the square root of both sides and isolating $x$. The solution to the equation is $x = -7.5 \pm \sqrt{34}$.

Example Problems

Here are some example problems that you can try to practice completing the square:

• $x^2 + 12x = 36$
• $x^2 + 20x = 100$
• $x^2 + 8x = 16$

Tips and Tricks

Here are some tips and tricks that you can use to help you complete the square:

• Make sure to move the constant term to the right-hand side of the equation.
• Find the square root of the coefficient of the $x$ term.
• Add and subtract the square of the square root to both sides of the equation.
• Simplify the equation to get it in the form $(x + a)^2 = b$.
• Take the square root of both sides of the equation to solve for $x$.

Common Mistakes

Here are some common mistakes that you can make when completing the square:

• Not moving the constant term to the right-hand side of the equation.
• Not finding the square root of the coefficient of the $x$ term.
• Not adding and subtracting the square of the square root to both sides of the equation.
• Not simplifying the equation to get it in the form $(x + a)^2 = b$.
• Not taking the square root of both sides of the equation to solve for $x$.

Conclusion

$$$$$$

Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves rewriting the equation in a perfect square form by adding and subtracting a constant term. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Write the equation in the form $ax^2 + bx + c = 0$.
2. Move the constant term to the right-hand side of the equation.
3. Find the square root of the coefficient of the $x$ term.
4. Add and subtract the square of the square root to both sides of the equation.
5. Simplify the equation to get it in the form $(x + a)^2 = b$.
6. Take the square root of both sides of the equation to solve for $x$.

Q: What is the square root of the coefficient of the $x$ term?

A: The square root of the coefficient of the $x$ term is the number that, when multiplied by itself, gives the coefficient of the $x$ term.

Q: How do I add and subtract the square of the square root to both sides of the equation?

A: To add and subtract the square of the square root to both sides of the equation, follow these steps:

1. Find the square of the square root by multiplying the square root by itself.
2. Add the square of the square root to both sides of the equation.
3. Subtract the square of the square root from both sides of the equation.

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

A: Completing the square and factoring are two different methods of solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves rewriting the equation in a perfect square form.

Q: When should I use completing the square?

A: You should use completing the square when the quadratic equation cannot be factored easily or when the equation is in the form $ax^2 + bx + c = 0$.

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 finding the square root of the coefficient of the $x$ term.
• Not adding and subtracting the square of the square root to both sides of the equation.
• Not simplifying the equation to get it in the form $(x + a)^2 = b$.
• Not taking the square root of both sides of the equation to solve for $x$.

Q: How do I check my work when completing the square?

A: To check your work when completing the square, follow these steps:

1. Rewrite the equation in the form $(x + a)^2 = b$.
2. Take the square root of both sides of the equation to solve for $x$.
3. Check that the solution satisfies the original equation.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. It involves rewriting the equation in a perfect square form by adding and subtracting a constant term. We have answered some frequently asked questions about completing the square and provided some tips and tricks to help you complete the square successfully.

Example Problems

Here are some example problems that you can try to practice completing the square:

• $x^2 + 12x = 36$
• $x^2 + 20x = 100$
• $x^2 + 8x = 16$

Tips and Tricks

Here are some tips and tricks that you can use to help you complete the square:

• Make sure to move the constant term to the right-hand side of the equation.
• Find the square root of the coefficient of the $x$ term.
• Add and subtract the square of the square root to both sides of the equation.
• Simplify the equation to get it in the form $(x + a)^2 = b$.
• Take the square root of both sides of the equation to solve for $x$.

Common Mistakes

Here are some common mistakes that you can make when completing the square:

• Not moving the constant term to the right-hand side of the equation.
• Not finding the square root of the coefficient of the $x$ term.
• Not adding and subtracting the square of the square root to both sides of the equation.
• Not simplifying the equation to get it in the form $(x + a)^2 = b$.
• Not taking the square root of both sides of the equation to solve for $x$.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. It involves rewriting the equation in a perfect square form by adding and subtracting a constant term. We have answered some frequently asked questions about completing the square and provided some tips and tricks to help you complete the square successfully.