Solve The Following Quadratic Function By Completing The Square.${ Y = X^2 + 10x + 10 }$ { X = [?] \pm \sqrt{\square} \}

Mar 10, 2025 by ADMIN 123 views

**Solving Quadratic Functions by Completing the Square**

Introduction

Quadratic functions are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. One of the methods used to solve quadratic functions is by completing the square. This method involves manipulating the quadratic function to express it in a perfect square form, which can then be easily solved. In this article, we will explore how to solve the quadratic function $y = x^2 + 10x + 10$ by completing the square.

What is Completing the Square?

Completing the square is a technique used to solve quadratic functions by expressing them in a perfect square form. This involves adding and subtracting a constant term to the quadratic function, which allows us to rewrite it in a form that can be easily solved. The constant term added is called the "square root" of the quadratic function, and it is used to create a perfect square trinomial.

Step 1: Write the Quadratic Function in Standard Form

The first step in completing the square is to write the quadratic function in standard form. The standard form of a quadratic function is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this case, the quadratic function is $y = x^2 + 10x + 10$ , which is already in standard form.

Step 2: Add and Subtract a Constant Term

The next step is to add and subtract a constant term to the quadratic function. This constant term is called the "square root" of the quadratic function, and it is used to create a perfect square trinomial. To find the square root, we need to take half of the coefficient of the $x$ term and square it. In this case, the coefficient of the $x$ term is 10, so we take half of 10, which is 5, and square it, which gives us 25.

y = x^2 + 10x + 10
y = (x^2 + 10x + 25) - 25 + 10
y = (x + 5)^2 - 15

Step 3: Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In this case, the perfect square trinomial is $(x + 5)^2$ , which can be factored as $(x + 5)(x + 5)$ .

y = (x + 5)^2 - 15
y = (x + 5)(x + 5) - 15

Step 4: Solve for x

The final step is to solve for $x$ . To do this, we need to set the expression equal to zero and solve for $x$ . In this case, we have:

(x + 5)(x + 5) - 15 = 0
(x + 5)^2 - 15 = 0
(x + 5)^2 = 15
x + 5 = ±√15
x = -5 ± √15

Conclusion

In this article, we have explored how to solve the quadratic function $y = x^2 + 10x + 10$ by completing the square. We have seen how to write the quadratic function in standard form, add and subtract a constant term, factor the perfect square trinomial, and solve for $x$ . By following these steps, we can easily solve quadratic functions using the method of completing the square.

Example Problems

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

Solve the quadratic function $y = x^2 + 6x + 8$ by completing the square.
Solve the quadratic function $y = x^2 - 4x + 4$ by completing the square.
Solve the quadratic function $y = x^2 + 2x + 1$ by completing the square.

Tips and Tricks

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

Make sure to take half of the coefficient of the $x$ term and square it to find the constant term.
Use the formula $(x + a)^2 = x^2 + 2ax + a^2$ to help you factor the perfect square trinomial.
Don't forget to add and subtract the constant term to the quadratic function.

Common Mistakes

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

Don't forget to take half of the coefficient of the $x$ term and square it to find the constant term.
Don't forget to add and subtract the constant term to the quadratic function.
Don't forget to factor the perfect square trinomial.

Real-World Applications

Completing the square has many real-world applications, including:

Physics: Completing the square is used to solve problems involving motion and energy.
Engineering: Completing the square is used to solve problems involving electrical circuits and mechanical systems.
Computer Science: Completing the square is used to solve problems involving algorithms and data structures.

Conclusion

Introduction

Completing the square is a powerful technique used to solve quadratic functions. In our previous article, we explored how to solve the quadratic function $y = x^2 + 10x + 10$ by completing the square. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic functions by expressing them in a perfect square form. This involves adding and subtracting a constant term to the quadratic function, which allows us to rewrite it in a form that can be easily solved.

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

A: You should use completing the square when you are given a quadratic function in the form $y = ax^2 + bx + c$ and you need to solve for $x$ . Completing the square is particularly useful when the quadratic function is not easily factorable.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

y = (x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c

Q: How do I find the constant term to add and subtract?

A: To find the constant term to add and subtract, you need to take half of the coefficient of the $x$ term and square it. In the formula above, the constant term is $-(\frac{b}{2})^2$ .

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

A: Completing the square and factoring are two different techniques used to solve quadratic functions. Factoring involves expressing the quadratic function as a product of two binomials, while completing the square involves expressing the quadratic function in a perfect square form.

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

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

Q: How do I know if a quadratic function can be solved by completing the square?

A: A quadratic function can be solved by completing the square if it can be expressed in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are real numbers. If the quadratic function has complex coefficients, it may not be possible to solve it by completing the square.

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

A: Yes, you can use completing the square to solve quadratic functions with rational coefficients. However, you will need to use rational arithmetic to solve the problem.

Q: How do I know if a quadratic function has a real solution?

A: A quadratic function has a real solution if the discriminant is non-negative. The discriminant is given by the formula $b^2 - 4ac$ .

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

A: Yes, you can use completing the square to solve quadratic functions with irrational coefficients. However, you will need to use irrational arithmetic to solve the problem.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic functions. By following the steps outlined in this article, you can easily solve quadratic functions using the method of completing the square. Remember to take half of the coefficient of the $x$ term and square it to find the constant term, use the formula $(x + a)^2 = x^2 + 2ax + a^2$ to help you factor the perfect square trinomial, and don't forget to add and subtract the constant term to the quadratic function. With practice and patience, you can become proficient in completing the square and solving quadratic functions.

Example Problems

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

Solve the quadratic function $y = x^2 + 6x + 8$ by completing the square.
Solve the quadratic function $y = x^2 - 4x + 4$ by completing the square.
Solve the quadratic function $y = x^2 + 2x + 1$ by completing the square.

Tips and Tricks

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

Make sure to take half of the coefficient of the $x$ term and square it to find the constant term.
Use the formula $(x + a)^2 = x^2 + 2ax + a^2$ to help you factor the perfect square trinomial.
Don't forget to add and subtract the constant term to the quadratic function.

Common Mistakes

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

Don't forget to take half of the coefficient of the $x$ term and square it to find the constant term.
Don't forget to add and subtract the constant term to the quadratic function.
Don't forget to factor the perfect square trinomial.

Real-World Applications

Completing the square has many real-world applications, including:

Physics: Completing the square is used to solve problems involving motion and energy.
Engineering: Completing the square is used to solve problems involving electrical circuits and mechanical systems.
Computer Science: Completing the square is used to solve problems involving algorithms and data structures.