Solve Each Quadratic Equation By Completing The Square. Simplify All Irrational Solutions.5. $x^2 - 6x - 16 = 0$6. $x^2 - 20x + 19 = 0$7. $x^2 + 4x + 14 = 46$8. X 2 + 14 X + 32 = − 8 X^2 + 14x + 32 = -8 X 2 + 14 X + 32 = − 8

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the methods for solving quadratic equations is by completing the square. This method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will explore how to solve quadratic equations by completing the square and simplify irrational solutions.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x + d)^2 = e, where d and e are constants. This form can be easily solved by taking the square root of both sides.

Step-by-Step Guide to Completing the Square

To complete the square, follow these steps:

1. Write the equation in the standard form: The equation should be in the form ax^2 + bx + c = 0.
2. Move the constant term to the right-hand side: Subtract c from both sides of the equation to isolate the quadratic term.
3. Divide the coefficient of the x-term by 2: Divide the coefficient of the x-term (b) by 2 and square the result.
4. Add the squared result to both sides: Add the squared result to both sides of the equation.
5. Write the left-hand side as a perfect square: The left-hand side of the equation should now be a perfect square trinomial.

Solving Quadratic Equations by Completing the Square

Example 1: $x^2 - 6x - 16 = 0$

To solve this equation by completing the square, follow the steps outlined above:

1. Write the equation in the standard form: $x^2 - 6x - 16 = 0$
2. Move the constant term to the right-hand side: $x^2 - 6x = 16$
3. Divide the coefficient of the x-term by 2: $-6/2 = -3$
4. Square the result: $(-3)^2 = 9$
5. Add the squared result to both sides: $x^2 - 6x + 9 = 16 + 9$
6. Write the left-hand side as a perfect square: $(x - 3)^2 = 25$

Now, take the square root of both sides:

$x - 3 = \pm \sqrt{25}$

$x - 3 = \pm 5$

$x = 3 \pm 5$

$x = 8$ or $x = -2$

Example 2: $x^2 - 20x + 19 = 0$

To solve this equation by completing the square, follow the steps outlined above:

1. Write the equation in the standard form: $x^2 - 20x + 19 = 0$
2. Move the constant term to the right-hand side: $x^2 - 20x = -19$
3. Divide the coefficient of the x-term by 2: $-20/2 = -10$
4. Square the result: $(-10)^2 = 100$
5. Add the squared result to both sides: $x^2 - 20x + 100 = -19 + 100$
6. Write the left-hand side as a perfect square: $(x - 10)^2 = 81$

Now, take the square root of both sides:

$x - 10 = \pm \sqrt{81}$

$x - 10 = \pm 9$

$x = 10 \pm 9$

$x = 19$ or $x = 1$

Example 3: $x^2 + 4x + 14 = 46$

To solve this equation by completing the square, follow the steps outlined above:

1. Write the equation in the standard form: $x^2 + 4x + 14 = 46$
2. Move the constant term to the right-hand side: $x^2 + 4x = 46 - 14$
3. Divide the coefficient of the x-term by 2: $4/2 = 2$
4. Square the result: $(2)^2 = 4$
5. Add the squared result to both sides: $x^2 + 4x + 4 = 46 - 14 + 4$
6. Write the left-hand side as a perfect square: $(x + 2)^2 = 36$

Now, take the square root of both sides:

$x + 2 = \pm \sqrt{36}$

$x + 2 = \pm 6$

$x = -2 \pm 6$

$x = 4$ or $x = -8$

Example 4: $x^2 + 14x + 32 = -8$

To solve this equation by completing the square, follow the steps outlined above:

1. Write the equation in the standard form: $x^2 + 14x + 32 = -8$
2. Move the constant term to the right-hand side: $x^2 + 14x = -8 - 32$
3. Divide the coefficient of the x-term by 2: $14/2 = 7$
4. Square the result: $(7)^2 = 49$
5. Add the squared result to both sides: $x^2 + 14x + 49 = -8 - 32 + 49$
6. Write the left-hand side as a perfect square: $(x + 7)^2 = 9$

Now, take the square root of both sides:

$x + 7 = \pm \sqrt{9}$

$x + 7 = \pm 3$

$x = -7 \pm 3$

$x = -4$ or $x = -10$

Conclusion

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x + d)^2 = e, where d and e are constants.

Q: How do I know if an equation can be solved by completing the square?

A: An equation can be solved by completing the square if it is in the form ax^2 + bx + c = 0, where a, b, and c are constants. If the equation is not in this form, you may need to use a different method to solve it.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

1. Write the equation in the standard form: ax^2 + bx + c = 0
2. Move the constant term to the right-hand side: ax^2 + bx = -c
3. Divide the coefficient of the x-term by 2: b/2
4. Square the result: (b/2)^2
5. Add the squared result to both sides: ax^2 + bx + (b/2)^2 = -c + (b/2)^2
6. Write the left-hand side as a perfect square: (x + b/2)^2 = -c + (b/2)^2

Q: How do I simplify irrational solutions?

A: To simplify irrational solutions, you can use the following steps:

1. Take the square root of both sides of the equation: x + b/2 = ±√(-c + (b/2)^2)
2. Simplify the expression under the square root: -c + (b/2)^2
3. Take the square root of the simplified expression: ±√(-c + (b/2)^2)
4. Simplify the expression: x = -b/2 ± √(-c + (b/2)^2)

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

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

• Not writing the equation in the standard form
• Not moving the constant term to the right-hand side
• Not dividing the coefficient of the x-term by 2
• Not squaring the result
• Not adding the squared result to both sides
• Not writing the left-hand side as a perfect square

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

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and follow the same steps as before.

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

A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. However, you will need to follow the same steps as before and simplify the expression under the square root.

Q: Is completing the square a reliable method for solving quadratic equations?

A: Yes, completing the square is a reliable method for solving quadratic equations. However, it may not always be the easiest method to use, especially for equations with complex coefficients or rational coefficients.

Q: Can I use completing the square to solve quadratic equations with multiple variables?

A: No, completing the square is typically used to solve quadratic equations with a single variable. If you have a quadratic equation with multiple variables, you may need to use a different method to solve it.

Conclusion

Solving quadratic equations by completing the square is a powerful technique that can be used to solve a wide range of equations. By following the steps outlined above and avoiding common mistakes, you can easily solve quadratic equations and simplify irrational solutions. Remember to always write the equation in the standard form, move the constant term to the right-hand side, divide the coefficient of the x-term by 2, square the result, add the squared result to both sides, and write the left-hand side as a perfect square. With practice, you will become proficient in solving quadratic equations by completing the square.