1) Use The Method Of Completing The Square To Solve The Equations.i) $x^2 - 3x - 7 = 0$ii) $p^2 + 5p + 3 = 0$

Introduction

In algebra, quadratic equations are a type of polynomial equation of degree two, which means the highest power of the variable is two. These equations are often written in the form of ax^2 + bx + c = 0, where a, b, and c are constants. One of the methods used to solve quadratic equations is the completing the square method. This method involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved.

What is Completing the Square?

Completing the square is a technique used to rewrite a quadratic equation in a form that allows us to easily find the solutions. The process involves adding and subtracting a constant term to create a perfect square trinomial. This constant term is called the "constant of completion." By adding and subtracting the same value, we are not changing the value of the equation, but we are making it easier to solve.

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 of ax^2 + bx + c = 0.
2. Move the constant term to the right-hand side: Subtract the constant term from both sides of the equation to isolate the terms with the variable.
3. Divide the coefficient of the variable term by 2: Divide the coefficient of the variable term (bx) by 2 to find the constant of completion.
4. Add the constant of completion to both sides: Add the constant of completion to both sides of the equation to create a perfect square trinomial.
5. Simplify the equation: Simplify the equation by combining like terms.

Solving the First Equation: $x^2 - 3x - 7 = 0$

To solve the equation $x^2 - 3x - 7 = 0$ using the completing the square method, follow these steps:

1. Write the equation in the standard form: The equation is already in the standard form.

2. Move the constant term to the right-hand side: Subtract -7 from both sides of the equation to isolate the terms with the variable.

   $x^2 - 3x = 7$

3. Divide the coefficient of the variable term by 2: Divide the coefficient of the variable term (-3) by 2 to find the constant of completion.

   $-\frac{3}{2}$

4. Add the constant of completion to both sides: Add the constant of completion to both sides of the equation to create a perfect square trinomial.

   $x^2 - 3x + \left(-\frac{3}{2}\right)^2 = 7 + \left(-\frac{3}{2}\right)^2$

5. Simplify the equation: Simplify the equation by combining like terms.

   $x^2 - 3x + \frac{9}{4} = 7 + \frac{9}{4}$ $x^2 - 3x + \frac{9}{4} = \frac{28}{4} + \frac{9}{4}$ $x^2 - 3x + \frac{9}{4} = \frac{37}{4}$ $\left(x - \frac{3}{2}\right)^2 = \frac{37}{4}$

6. Take the square root of both sides: Take the square root of both sides of the equation to find the solutions.

   $x - \frac{3}{2} = \pm \sqrt{\frac{37}{4}}$ $x - \frac{3}{2} = \pm \frac{\sqrt{37}}{2}$ $x = \frac{3}{2} \pm \frac{\sqrt{37}}{2}$

Solving the Second Equation: $p^2 + 5p + 3 = 0$

To solve the equation $p^2 + 5p + 3 = 0$ using the completing the square method, follow these steps:

1. Write the equation in the standard form: The equation is already in the standard form.

2. Move the constant term to the right-hand side: Subtract 3 from both sides of the equation to isolate the terms with the variable.

   $p^2 + 5p = -3$

3. Divide the coefficient of the variable term by 2: Divide the coefficient of the variable term (5) by 2 to find the constant of completion.

   $\frac{5}{2}$

4. Add the constant of completion to both sides: Add the constant of completion to both sides of the equation to create a perfect square trinomial.

   $p^2 + 5p + \left(\frac{5}{2}\right)^2 = -3 + \left(\frac{5}{2}\right)^2$

5. Simplify the equation: Simplify the equation by combining like terms.

   $p^2 + 5p + \frac{25}{4} = -3 + \frac{25}{4}$ $p^2 + 5p + \frac{25}{4} = \frac{-12 + 25}{4}$ $p^2 + 5p + \frac{25}{4} = \frac{13}{4}$ $\left(p + \frac{5}{2}\right)^2 = \frac{13}{4}$

6. Take the square root of both sides: Take the square root of both sides of the equation to find the solutions.

   $p + \frac{5}{2} = \pm \sqrt{\frac{13}{4}}$ $p + \frac{5}{2} = \pm \frac{\sqrt{13}}{2}$ $p = -\frac{5}{2} \pm \frac{\sqrt{13}}{2}$

Conclusion

Introduction

In the previous article, we discussed the completing the square method to solve quadratic equations. This method involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved. In this article, we will answer some frequently asked questions about the completing the square method.

Q: What is the completing the square method?

A: The completing the square method is a technique used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form.

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

A: You should use the completing the square method when you are given a quadratic equation in the form of ax^2 + bx + c = 0, and you want to find the solutions.

Q: What are the steps involved in the completing the square method?

A: The steps involved in the completing the square method are:

1. Write the equation in the standard form: The equation should be in the form of ax^2 + bx + c = 0.
2. Move the constant term to the right-hand side: Subtract the constant term from both sides of the equation to isolate the terms with the variable.
3. Divide the coefficient of the variable term by 2: Divide the coefficient of the variable term (bx) by 2 to find the constant of completion.
4. Add the constant of completion to both sides: Add the constant of completion to both sides of the equation to create a perfect square trinomial.
5. Simplify the equation: Simplify the equation by combining like terms.

Q: What is the constant of completion?

A: The constant of completion is the value that is added to both sides of the equation to create a perfect square trinomial. It is found by dividing the coefficient of the variable term by 2.

Q: How do I find the solutions to the equation?

A: To find the solutions to the equation, take the square root of both sides of the equation. This will give you two possible solutions.

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

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

• Not writing the equation in the standard form: Make sure the equation is in the form of ax^2 + bx + c = 0.
• Not moving the constant term to the right-hand side: Subtract the constant term from both sides of the equation to isolate the terms with the variable.
• Not dividing the coefficient of the variable term by 2: Divide the coefficient of the variable term (bx) by 2 to find the constant of completion.
• Not adding the constant of completion to both sides: Add the constant of completion to both sides of the equation to create a perfect square trinomial.
• Not simplifying the equation: Simplify the equation by combining like terms.

Q: Can the completing the square method be used to solve all quadratic equations?

A: No, the completing the square method cannot be used to solve all quadratic equations. This method is only used to solve quadratic equations that can be expressed in the form of ax^2 + bx + c = 0.

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

A: The completing the square method has many real-world applications, including:

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

Conclusion

In this article, we have answered some frequently asked questions about the completing the square method. This method is a powerful tool for solving quadratic equations and has many real-world applications. By following the steps outlined in this article, you can use the completing the square method to solve quadratic equations and find the solutions.