Select The Correct Answer.Solve The Equation By Completing The Square: $0 = 4x^2 - 72x$A. $x = 0, 18$ B. $x = -18, 0$ C. $x = -72, 90$ D. $x = -90, 72$

Introduction

In algebra, completing the square is a method used to solve quadratic equations. This method involves rewriting the equation in a perfect square trinomial form, which allows us to easily find the solutions. In this article, we will solve the equation $0 = 4x^2 - 72x$ by completing the square.

Step 1: Write the Equation in General Form

The given equation is $0 = 4x^2 - 72x$. To complete the square, we need to write the equation in the general form $ax^2 + bx + c = 0$. In this case, $a = 4$, $b = -72$, and $c = 0$.

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 4. This gives us $0 = x^2 - 18x$.

Step 3: Move the Constant Term to the Right-Hand Side

Now, we need to move the constant term to the right-hand side of the equation. This gives us $x^2 - 18x = 0$.

Step 4: Add and Subtract the Square of Half the Coefficient of $x$

To complete the square, we need to add and subtract the square of half the coefficient of $x$. Half of the coefficient of $x$ is $-18/2 = -9$. The square of $-9$ is $81$. So, we add and subtract 81 to the equation: $x^2 - 18x + 81 - 81 = 0$.

Step 5: Factor the Perfect Square Trinomial

The equation can now be factored as a perfect square trinomial: $(x - 9)^2 - 81 = 0$.

Step 6: Add 81 to Both Sides

To isolate the perfect square trinomial, we need to add 81 to both sides of the equation: $(x - 9)^2 = 81$.

Step 7: Take the Square Root of Both Sides

Now, we can take the square root of both sides of the equation: $x - 9 = \pm \sqrt{81}$.

Step 8: Simplify the Square Root

The square root of 81 is 9. So, we can simplify the equation as $x - 9 = \pm 9$.

Step 9: Add 9 to Both Sides

To solve for $x$, we need to add 9 to both sides of the equation: $x = 9 \pm 9$.

Step 10: Simplify the Solutions

The solutions to the equation are $x = 9 + 9 = 18$ and $x = 9 - 9 = 0$.

Conclusion

In this article, we solved the equation $0 = 4x^2 - 72x$ by completing the square. We wrote the equation in general form, divided it by the coefficient of $x^2$, moved the constant term to the right-hand side, added and subtracted the square of half the coefficient of $x$, factored the perfect square trinomial, added 81 to both sides, took the square root of both sides, simplified the square root, added 9 to both sides, and simplified the solutions. The solutions to the equation are $x = 0, 18$.

Answer

The correct answer is A. $x = 0, 18$.

Discussion

This method of solving quadratic equations by completing the square is a powerful tool in algebra. It allows us to easily find the solutions to quadratic equations, even when the equations are not in the standard form. By following the steps outlined in this article, we can solve any quadratic equation by completing the square.

Example Problems

1. Solve the equation $0 = x^2 + 6x$ by completing the square.
2. Solve the equation $0 = 2x^2 - 12x$ by completing the square.
3. Solve the equation $0 = x^2 - 10x$ by completing the square.

Solutions

1. The solutions to the equation $0 = x^2 + 6x$ are $x = -3, 0$.
2. The solutions to the equation $0 = 2x^2 - 12x$ are $x = 0, 6$.
3. The solutions to the equation $0 = x^2 - 10x$ are $x = 0, 10$.

Conclusion

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Introduction

Completing the square is a method used to solve quadratic equations. It involves rewriting the equation in a perfect square trinomial form, which allows us to easily find the solutions. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a method used to solve quadratic equations by rewriting the equation in a perfect square trinomial form.

Q: Why do we need to complete the square?

A: We need to complete the square because it allows us to easily find the solutions to quadratic equations. By rewriting the equation in a perfect square trinomial form, we can use the square root property to find the solutions.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

1. Write the equation in general form.
2. Divide the equation by the coefficient of $x^2$.
3. Move the constant term to the right-hand side.
4. Add and subtract the square of half the coefficient of $x$.
5. Factor the perfect square trinomial.
6. Add 81 to both sides.
7. Take the square root of both sides.
8. Simplify the square root.
9. Add 9 to both sides.
10. Simplify the solutions.

Q: What is the square root property?

A: The square root property states that if $x^2 = a$, then $x = \pm \sqrt{a}$.

Q: How do I use the square root property to find the solutions?

A: To use the square root property, you need to take the square root of both sides of the equation. This will give you two possible solutions: $x = \pm \sqrt{a}$.

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

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

• Not dividing the equation by the coefficient of $x^2$.
• Not moving the constant term to the right-hand side.
• Not adding and subtracting the square of half the coefficient of $x$.
• Not factoring the perfect square trinomial.
• Not adding 81 to both sides.
• Not taking the square root of both sides.
• Not simplifying the square root.
• Not adding 9 to both sides.
• Not simplifying the solutions.

Q: Can I use completing the square to solve any quadratic equation?

A: Yes, you can use completing the square to solve any quadratic equation. However, you need to make sure that the equation is in the general form $ax^2 + bx + c = 0$.

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

A: Some real-world applications of completing the square include:

• Solving quadratic equations in physics and engineering.
• Finding the maximum or minimum value of a quadratic function.
• Solving systems of quadratic equations.
• Finding the roots of a quadratic equation.

Conclusion

In conclusion, completing the square is a powerful method for solving quadratic equations. By following the steps outlined in this article, you can easily find the solutions to quadratic equations. Remember to avoid common mistakes and use the square root property to find the solutions.

Frequently Asked Questions

1. Q: What is the difference between completing the square and factoring? A: Completing the square involves rewriting the equation in a perfect square trinomial form, while factoring involves expressing the equation as a product of two binomials.
2. Q: Can I use completing the square to solve a quadratic equation with a negative coefficient of $x^2$? A: Yes, you can use completing the square to solve a quadratic equation with a negative coefficient of $x^2$. However, you need to make sure that the equation is in the general form $ax^2 + bx + c = 0$.
3. Q: How do I know if an equation can be solved by completing the square? A: You can check if an equation can be solved by completing the square by looking at the coefficient of $x^2$. If the coefficient of $x^2$ is not equal to 1, you need to divide the equation by the coefficient of $x^2$ before completing the square.

Example Problems

1. Solve the equation $0 = x^2 + 6x$ by completing the square.
2. Solve the equation $0 = 2x^2 - 12x$ by completing the square.
3. Solve the equation $0 = x^2 - 10x$ by completing the square.

Solutions

1. The solutions to the equation $0 = x^2 + 6x$ are $x = -3, 0$.
2. The solutions to the equation $0 = 2x^2 - 12x$ are $x = 0, 6$.
3. The solutions to the equation $0 = x^2 - 10x$ are $x = 0, 10$.