Solve The Equation Using The Method Of Completing The Square.$\[ 2x^2 + 16x - 8 = 0 \\]A. $\[ X = -4 \pm 2\sqrt{5} \\]B. $\[ X = -2 \pm 4\sqrt{5} \\]C. $\[ X = 4 \pm 2\sqrt{5} \\]D. $\[ X = 2 \pm 4\sqrt{5} \\]
Introduction
Quadratic equations 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 equations is the method of completing the square. This method involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will explore how to solve the equation 2x^2 + 16x - 8 = 0 using the method of completing the square.
Understanding the Method of Completing the Square
The method of completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. The basic idea behind this method is to manipulate the equation to express it in the form (x + d)^2 = e, where d and e are constants. This can be done by adding and subtracting a constant term to the equation, which allows us to create a perfect square trinomial.
Step 1: Write Down the Given Equation
The given equation is 2x^2 + 16x - 8 = 0. We will use this equation to demonstrate the method of completing the square.
Step 2: Divide the Equation by the Coefficient of x^2
To make the coefficient of x^2 equal to 1, we need to divide the entire equation by 2. This gives us:
x^2 + 8x - 4 = 0
Step 3: Add and Subtract a Constant Term
To create a perfect square trinomial, we need to add and subtract a constant term to the equation. The constant term we need to add is (8/2)^2 = 16. However, since we are adding 16, we need to subtract 16 as well to keep the equation balanced. This gives us:
x^2 + 8x + 16 - 16 - 4 = 0
Step 4: Simplify the Equation
We can simplify the equation by combining like terms:
(x^2 + 8x + 16) - 20 = 0
Step 5: Express the Perfect Square Trinomial
The expression x^2 + 8x + 16 is a perfect square trinomial, which can be expressed as (x + 4)^2. Therefore, we can rewrite the equation as:
(x + 4)^2 - 20 = 0
Step 6: Add 20 to Both Sides
To isolate the perfect square trinomial, we need to add 20 to both sides of the equation:
(x + 4)^2 = 20
Step 7: Take the Square Root of Both Sides
To solve for x, we need to take the square root of both sides of the equation:
x + 4 = ±√20
Step 8: Simplify the Square Root
We can simplify the square root by expressing it as √(4*5) = 2√5. Therefore, we can rewrite the equation as:
x + 4 = ±2√5
Step 9: Solve for x
To solve for x, we need to subtract 4 from both sides of the equation:
x = -4 ± 2√5
Conclusion
In this article, we have demonstrated how to solve the equation 2x^2 + 16x - 8 = 0 using the method of completing the square. We have shown that the solution to the equation is x = -4 ± 2√5. This method is a powerful tool for solving quadratic equations, and it can be used to solve a wide range of equations.
Comparison with the Given Options
The solution we obtained is x = -4 ± 2√5. Let's compare this with the given options:
A. x = -4 ± 2√5 B. x = -2 ± 4√5 C. x = 4 ± 2√5 D. x = 2 ± 4√5
The correct solution is option A, which is x = -4 ± 2√5.
Final Answer
Q: What is the method of completing the square?
A: The method of 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: What are the steps involved in solving a quadratic equation using the method of completing the square?
A: The steps involved in solving a quadratic equation using the method of completing the square are:
- Write down the given equation.
- Divide the equation by the coefficient of x^2.
- Add and subtract a constant term to create a perfect square trinomial.
- Simplify the equation.
- Express the perfect square trinomial.
- Add the constant term to both sides of the equation.
- Take the square root of both sides of the equation.
- Solve for x.
Q: What is the purpose of adding and subtracting a constant term in the method of completing the square?
A: The purpose of adding and subtracting a constant term is to create a perfect square trinomial. This allows us to express the equation in the form (x + d)^2 = e, where d and e are constants.
Q: How do I know when to add and subtract a constant term?
A: You know when to add and subtract a constant term when you see a quadratic expression that can be written as a perfect square trinomial. For example, if you have the expression x^2 + 8x + 16, you can add and subtract 16 to create a perfect square trinomial.
Q: What is the difference between the method of completing the square and factoring?
A: The method of completing the square and factoring are two different techniques used to solve quadratic equations. Factoring involves expressing the equation as a product of two binomials, while the method of completing the square involves manipulating the equation to express it in the form (x + d)^2 = e.
Q: When should I use the method of completing the square?
A: You should use the method of completing the square when you have a quadratic equation that cannot be easily factored. This method is particularly useful when the equation has a coefficient of 1 for the x^2 term.
Q: Can I use the method of completing the square to solve quadratic equations with complex coefficients?
A: Yes, you can use the method of completing the square to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and complex arithmetic to solve the equation.
Q: How do I check my solution to a quadratic equation using the method of completing the square?
A: To check your solution, you can plug the value of x back into the original equation and simplify. If the equation is true, then your solution is correct.
Conclusion
In this article, we have answered some of the most frequently asked questions about solving quadratic equations using the method of completing the square. We have covered topics such as the steps involved in solving a quadratic equation, the purpose of adding and subtracting a constant term, and when to use the method of completing the square. We hope that this article has been helpful in clarifying any confusion you may have had about this topic.