ExercisesSolve Each Equation By Completing The Square. Round To The Nearest Tenth If Necessary.1. $x^2 - 4x + 3 = 0$2. $x^2 + 10x = -9$3. $x^2 - 8x - 9 = 0$4. $x^2 - 6x = 16$5. $x^2 - 4x - 5 = 0$6. $x^2

What is Completing the Square?

Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. This method involves manipulating the equation to express it in the form $(x + d)^2 = e$, where $d$ and $e$ are constants. By doing so, we can easily find the solutions to the equation.

Step-by-Step Guide to Completing the Square

To complete the square, follow these steps:

1. Write the equation in the form $ax^2 + bx + c = 0$: Make sure the equation is in the standard form of a quadratic equation.
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 $b$ by 2 to get $\frac{b}{2}$.
4. Square the result: Square $\frac{b}{2}$ to get $\left(\frac{b}{2}\right)^2$.
5. Add the squared result to both sides: Add $\left(\frac{b}{2}\right)^2$ to both sides of the equation.
6. Write the left-hand side as a perfect square: The left-hand side of the equation should now be a perfect square, in the form $(x + d)^2$.

Solving Each Equation by Completing the Square

1. $x^2 - 4x + 3 = 0$

To solve this equation, we will follow the steps outlined above.

• Write the equation in the form $ax^2 + bx + c = 0$: The equation is already in the standard form.
• Move the constant term to the right-hand side: Subtract 3 from both sides of the equation to get $x^2 - 4x = -3$.
• Divide the coefficient of the $x$ term by 2: Divide -4 by 2 to get -2.
• Square the result: Square -2 to get 4.
• Add the squared result to both sides: Add 4 to both sides of the equation to get $x^2 - 4x + 4 = 1$.
• Write the left-hand side as a perfect square: The left-hand side of the equation is now a perfect square, in the form $(x - 2)^2$.

The equation can now be written as $(x - 2)^2 = 1$. To solve for $x$, take the square root of both sides of the equation.

$$x - 2 = \pm \sqrt{1}$$

Simplifying the equation, we get:

$$x - 2 = \pm 1$$

Adding 2 to both sides of the equation, we get:

$$x = 3 \text{ or } x = 1$$

2. $x^2 + 10x = -9$

To solve this equation, we will follow the steps outlined above.

• Write the equation in the form $ax^2 + bx + c = 0$: Add 9 to both sides of the equation to get $x^2 + 10x + 9 = 0$.
• Move the constant term to the right-hand side: The equation is already in the standard form.
• Divide the coefficient of the $x$ term by 2: Divide 10 by 2 to get 5.
• Square the result: Square 5 to get 25.
• Add the squared result to both sides: Add 25 to both sides of the equation to get $x^2 + 10x + 25 = 16$.
• Write the left-hand side as a perfect square: The left-hand side of the equation is now a perfect square, in the form $(x + 5)^2$.

The equation can now be written as $(x + 5)^2 = 16$. To solve for $x$, take the square root of both sides of the equation.

$$x + 5 = \pm \sqrt{16}$$

Simplifying the equation, we get:

$$x + 5 = \pm 4$$

Subtracting 5 from both sides of the equation, we get:

$$x = -1 \text{ or } x = -9$$

3. $x^2 - 8x - 9 = 0$

To solve this equation, we will follow the steps outlined above.

• Write the equation in the form $ax^2 + bx + c = 0$: The equation is already in the standard form.
• Move the constant term to the right-hand side: Add 9 to both sides of the equation to get $x^2 - 8x = 9$.
• Divide the coefficient of the $x$ term by 2: Divide -8 by 2 to get -4.
• Square the result: Square -4 to get 16.
• Add the squared result to both sides: Add 16 to both sides of the equation to get $x^2 - 8x + 16 = 25$.
• Write the left-hand side as a perfect square: The left-hand side of the equation is now a perfect square, in the form $(x - 4)^2$.

The equation can now be written as $(x - 4)^2 = 25$. To solve for $x$, take the square root of both sides of the equation.

$$x - 4 = \pm \sqrt{25}$$

Simplifying the equation, we get:

$$x - 4 = \pm 5$$

Adding 4 to both sides of the equation, we get:

$$x = 9 \text{ or } x = -1$$

4. $x^2 - 6x = 16$

To solve this equation, we will follow the steps outlined above.

• Write the equation in the form $ax^2 + bx + c = 0$: Add 16 to both sides of the equation to get $x^2 - 6x + 16 = 0$.
• Move the constant term to the right-hand side: The equation is already in the standard form.
• Divide the coefficient of the $x$ term by 2: Divide -6 by 2 to get -3.
• Square the result: Square -3 to get 9.
• Add the squared result to both sides: Add 9 to both sides of the equation to get $x^2 - 6x + 9 = 25$.
• Write the left-hand side as a perfect square: The left-hand side of the equation is now a perfect square, in the form $(x - 3)^2$.

The equation can now be written as $(x - 3)^2 = 25$. To solve for $x$, take the square root of both sides of the equation.

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

Simplifying the equation, we get:

$$x - 3 = \pm 5$$

Adding 3 to both sides of the equation, we get:

$$x = 8 \text{ or } x = -2$$

5. $x^2 - 4x - 5 = 0$

To solve this equation, we will follow the steps outlined above.

• Write the equation in the form $ax^2 + bx + c = 0$: The equation is already in the standard form.
• Move the constant term to the right-hand side: Add 5 to both sides of the equation to get $x^2 - 4x = 5$.
• Divide the coefficient of the $x$ term by 2: Divide -4 by 2 to get -2.
• Square the result: Square -2 to get 4.
• Add the squared result to both sides: Add 4 to both sides of the equation to get $x^2 - 4x + 4 = 9$.
• Write the left-hand side as a perfect square: The left-hand side of the equation is now a perfect square, in the form $(x - 2)^2$.

The equation can now be written as $(x - 2)^2 = 9$. To solve for $x$, take the square root of both sides of the equation.

$$x - 2 = \pm \sqrt{9}$$

Simplifying the equation, we get:

$$x - 2 = \pm 3$$

Adding 2 to both sides of the equation, we get:

$$x = 5 \text{ or } x = -1$$

6. $x^2 + 2x + 1 = 0$

To solve this equation, we will follow the steps outlined above.

• Write the equation in the form $ax^2 + bx + c = 0$: The equation is already in the standard form.
• Move the constant term to the right-hand side: Subtract 1 from both sides of the equation to get $x^2 + 2x = -1$.
• **Divide the coefficient
  Frequently Asked Questions (FAQs) about Solving Quadratic Equations by Completing the Square =============================================================================================

Q: What is completing the square?

A: Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. This method 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 form $ax^2 + bx + c = 0$: Make sure the equation is in the standard form of a quadratic equation.
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 $b$ by 2 to get $\frac{b}{2}$.
4. Square the result: Square $\frac{b}{2}$ to get $\left(\frac{b}{2}\right)^2$.
5. Add the squared result to both sides: Add $\left(\frac{b}{2}\right)^2$ to both sides of the equation.
6. Write the left-hand side as a perfect square: The left-hand side of the equation should now be a perfect square, in the form $(x + d)^2$.

Q: How do I solve the equation once I have completed the square?

A: Once you have completed the square, you can solve the equation by taking the square root of both sides of the equation. This will give you two possible solutions for $x$.

Q: What if the equation has no real solutions?

A: If the equation has no real solutions, it means that the equation has complex solutions. In this case, you can use the quadratic formula to find the complex solutions.

Q: Can I use completing the square to solve all types of quadratic equations?

A: Yes, you can use completing the square to solve all types of quadratic equations, including those with complex solutions.

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: Make sure to subtract $c$ from both sides of the equation to isolate the quadratic term.
• Not dividing the coefficient of the $x$ term by 2: Make sure to divide $b$ by 2 to get $\frac{b}{2}$.
• Not squaring the result: Make sure to square $\frac{b}{2}$ to get $\left(\frac{b}{2}\right)^2$.
• Not adding the squared result to both sides: Make sure to add $\left(\frac{b}{2}\right)^2$ to both sides of the equation.

Q: How do I know if I have completed the square correctly?

A: You can check if you have completed the square correctly by looking at the left-hand side of the equation. If it is in the form $(x + d)^2$, then you have completed the square correctly.

Q: What are some real-world applications of completing the square?

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

• Physics: Completing the square is used to solve equations of motion and to find the position and velocity of an object.
• Engineering: Completing the square is used to solve equations of electrical circuits and to find the current and voltage of a circuit.
• Computer Science: Completing the square is used to solve equations of algorithms and to find the time and space complexity of an algorithm.

Q: Can I use completing the square to solve systems of equations?

A: Yes, you can use completing the square to solve systems of equations. This involves completing the square for each equation in the system and then solving the resulting equations.

Q: What are some tips for mastering completing the square?

A: Some tips for mastering completing the square include:

• Practice, practice, practice: The more you practice completing the square, the more comfortable you will become with the method.
• Start with simple equations: Begin with simple equations and gradually move on to more complex ones.
• Use visual aids: Use visual aids such as graphs and charts to help you understand the concept of completing the square.
• Check your work: Make sure to check your work carefully to avoid making mistakes.