What Number Should Be Added To Both Sides Of The Equation To Complete The Square?$\[ X^2 + 3x = 6 \\]A. \[$\frac{3}{2}\$\]B. \[$\left(\frac{3}{2}\right)^2\$\]C. 3D. \[$6^2\$\]
What Number Should Be Added to Both Sides of the Equation to Complete the Square?
Understanding the Concept of Completing the Square
Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method involves manipulating the equation to create a perfect square trinomial on one side of the equation. The process of completing the square is essential in algebra and is used to solve quadratic equations that cannot be factored easily.
The General Form of a Quadratic Equation
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
Completing the Square: A Step-by-Step Guide
To complete the square, we need to follow these steps:
- Move the constant term to the right-hand side: We move the constant term (c) to the right-hand side of the equation by subtracting it from both sides.
- Divide the coefficient of the x-term by 2: We divide the coefficient of the x-term (b) by 2 and then square the result.
- Add the squared result to both sides: We add the squared result to both sides of the equation.
- Write the left-hand side as a perfect square trinomial: We write the left-hand side of the equation as a perfect square trinomial.
Applying the Concept to the Given Equation
Now, let's apply the concept of completing the square to the given equation:
x^2 + 3x = 6
Step 1: Move the constant term to the right-hand side
x^2 + 3x - 6 = 0
Step 2: Divide the coefficient of the x-term by 2
b = 3 b/2 = 3/2
Step 3: Square the result
(3/2)^2 = 9/4
Step 4: Add the squared result to both sides
x^2 + 3x + 9/4 = 6 + 9/4
Simplifying the Equation
To simplify the equation, we need to find a common denominator for the two terms on the right-hand side. The common denominator is 4.
x^2 + 3x + 9/4 = 24/4 + 9/4
Combine the two terms on the right-hand side:
x^2 + 3x + 9/4 = 33/4
Writing the Left-Hand Side as a Perfect Square Trinomial
Now, we can write the left-hand side of the equation as a perfect square trinomial:
(x + 3/2)^2 = 33/4
The Number to Be Added to Both Sides
To complete the square, we need to add the squared result (9/4) to both sides of the equation. Therefore, the number to be added to both sides of the equation is:
(3/2)^2 = 9/4
However, the options provided are:
A. 3/2 B. (3/2)^2 C. 3 D. 6^2
Since (3/2)^2 is the correct answer, we can eliminate options A, C, and D.
Conclusion
In conclusion, the number to be added to both sides of the equation to complete the square is (3/2)^2. This is because we need to add the squared result (9/4) to both sides of the equation to write the left-hand side as a perfect square trinomial.
Final Answer
The final answer is B. (3/2)^2.
Frequently Asked Questions (FAQs) About Completing the Square
Q: What is completing the square?
A: Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method involves manipulating the equation to create a perfect square trinomial on one side of the equation.
Q: Why is completing the square important?
A: Completing the square is essential in algebra and is used to solve quadratic equations that cannot be factored easily. It is also used to find the vertex of a parabola and to solve systems of equations.
Q: What is the general form of a quadratic equation?
A: The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
Q: How do I complete the square?
A: To complete the square, you need to follow these steps:
- Move the constant term to the right-hand side: We move the constant term (c) to the right-hand side of the equation by subtracting it from both sides.
- Divide the coefficient of the x-term by 2: We divide the coefficient of the x-term (b) by 2 and then square the result.
- Add the squared result to both sides: We add the squared result to both sides of the equation.
- Write the left-hand side as a perfect square trinomial: We write the left-hand side of the equation as a perfect square trinomial.
Q: What is the number to be added to both sides of the equation to complete the square?
A: The number to be added to both sides of the equation to complete the square is the squared result of the coefficient of the x-term divided by 2.
Q: How do I find the number to be added to both sides of the equation?
A: To find the number to be added to both sides of the equation, you need to follow these steps:
- Divide the coefficient of the x-term by 2: We divide the coefficient of the x-term (b) by 2.
- Square the result: We square the result of the division.
- Add the squared result to both sides: We add the squared result to both sides of the equation.
Q: What is the final form of the equation after completing the square?
A: The final form of the equation after completing the square is:
(x + b/2)^2 = c + (b/2)^2
where a, b, and c are constants, and x is the variable.
Q: How do I use completing the square to solve quadratic equations?
A: To use completing the square to solve quadratic equations, you need to follow these steps:
- Complete the square: We complete the square by following the steps outlined above.
- Solve for x: We solve for x by taking the square root of both sides of the equation.
Q: What are the advantages of completing the square?
A: The advantages of completing the square include:
- It is a powerful tool for solving quadratic equations.
- It is used to find the vertex of a parabola.
- It is used to solve systems of equations.
Q: What are the disadvantages of completing the square?
A: The disadvantages of completing the square include:
- It can be time-consuming.
- It requires a good understanding of algebra.
Conclusion
In conclusion, completing the square is a powerful tool for solving quadratic equations. It is used to find the vertex of a parabola and to solve systems of equations. While it can be time-consuming, it is a valuable technique to have in your mathematical toolkit.
Final Answer
The final answer is B. (3/2)^2.