For Which Equation Does It Make The Most Sense To Solve By Completing The Square?A. X 2 + 20 X = 52 X^2 + 20x = 52 X 2 + 20 X = 52 B. 5 X 2 + 3 X = 9 5x^2 + 3x = 9 5 X 2 + 3 X = 9 C. 3 X 2 − X + 17 = 0 3x^2 - X + 17 = 0 3 X 2 − X + 17 = 0 D. X 2 − 8 = 1 X^2 - 8 = 1 X 2 − 8 = 1

For Which Equation Does it Make the Most Sense to Solve by Completing the Square?

Completing the square is a powerful algebraic technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. However, not all quadratic equations are suitable for completing the square. In this article, we will examine four different quadratic equations and determine which one makes the most sense to solve by completing the square.

Completing the square is a method of solving quadratic equations of the form $ax^2 + bx + c = 0$. The process involves manipulating the equation to express it in the form $(x + p)^2 = q$, where $p$ and $q$ 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.

Equation A: $x^2 + 20x = 52$

Let's start by examining Equation A: $x^2 + 20x = 52$. To solve this equation by completing the square, we need to move the constant term to the right-hand side of the equation. This gives us:

$$x^2 + 20x - 52 = 0$$

Next, we need to find the value of $p$ that will allow us to create a perfect square trinomial. We can do this by taking half of the coefficient of the $x$ term and squaring it. In this case, the coefficient of the $x$ term is 20, so we take half of that and square it:

$$p = \frac{20}{2} = 10$$

$$p^2 = 10^2 = 100$$

Now, we can add $p^2$ to both sides of the equation to create a perfect square trinomial:

$$x^2 + 20x + 100 = 52 + 100$$

This simplifies to:

$$(x + 10)^2 = 152$$

Taking the square root of both sides gives us:

$$x + 10 = \pm \sqrt{152}$$

Subtracting 10 from both sides gives us the final solution:

$$x = -10 \pm \sqrt{152}$$

Equation B: $5x^2 + 3x = 9$

Now, let's examine Equation B: $5x^2 + 3x = 9$. To solve this equation by completing the square, we need to move the constant term to the right-hand side of the equation. This gives us:

$$5x^2 + 3x - 9 = 0$$

Next, we need to find the value of $p$ that will allow us to create a perfect square trinomial. We can do this by taking half of the coefficient of the $x$ term and squaring it. In this case, the coefficient of the $x$ term is 3, so we take half of that and square it:

$$p = \frac{3}{2}$$

$$p^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$

However, we also need to multiply the entire equation by 4 to get rid of the fraction, which will give us:

$$20x^2 + 12x - 36 = 0$$

Now, we can add $p^2$ to both sides of the equation to create a perfect square trinomial:

$$20x^2 + 12x + 9 = 36 + 9$$

This simplifies to:

$$(2x + \frac{3}{2})^2 = 45$$

However, we can simplify this further by multiplying both sides by 4 to get rid of the fraction:

$$(4x + 3)^2 = 180$$

Taking the square root of both sides gives us:

$$4x + 3 = \pm \sqrt{180}$$

Subtracting 3 from both sides gives us:

$$4x = -3 \pm \sqrt{180}$$

Dividing both sides by 4 gives us the final solution:

$$x = -\frac{3}{4} \pm \frac{\sqrt{180}}{4}$$

Equation C: $3x^2 - x + 17 = 0$

Now, let's examine Equation C: $3x^2 - x + 17 = 0$. To solve this equation by completing the square, we need to find the value of $p$ that will allow us to create a perfect square trinomial. However, in this case, the coefficient of the $x$ term is not a multiple of the coefficient of the $x^2$ term, so we cannot simply take half of the coefficient of the $x$ term and square it.

In this case, we need to use a different approach. We can start by dividing the entire equation by the coefficient of the $x^2$ term, which is 3:

$$x^2 - \frac{1}{3}x + \frac{17}{3} = 0$$

Next, we can add and subtract a constant term to the equation to create a perfect square trinomial. We can do this by adding and subtracting $\left(\frac{1}{6}\right)^2$ to the equation:

$$x^2 - \frac{1}{3}x + \left(\frac{1}{6}\right)^2 - \left(\frac{1}{6}\right)^2 + \frac{17}{3} = 0$$

This simplifies to:

$$\left(x - \frac{1}{6}\right)^2 - \left(\frac{1}{6}\right)^2 + \frac{17}{3} = 0$$

Adding $\left(\frac{1}{6}\right)^2$ to both sides of the equation gives us:

$$\left(x - \frac{1}{6}\right)^2 = \left(\frac{1}{6}\right)^2 - \frac{17}{3}$$

However, this is not a perfect square trinomial, so we cannot solve the equation by completing the square.

Equation D: $x^2 - 8 = 1$

Finally, let's examine Equation D: $x^2 - 8 = 1$. To solve this equation by completing the square, we need to move the constant term to the right-hand side of the equation. This gives us:

$$x^2 = 9$$

Taking the square root of both sides gives us:

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

Simplifying the square root gives us the final solution:

$$x = \pm 3$$

In conclusion, we have examined four different quadratic equations and determined which one makes the most sense to solve by completing the square. Equation A: $x^2 + 20x = 52$ is the best candidate for solving by completing the square, as it can be easily manipulated into a perfect square trinomial. Equation B: $5x^2 + 3x = 9$ can also be solved by completing the square, but it requires more manipulation than Equation A. Equation C: $3x^2 - x + 17 = 0$ cannot be solved by completing the square, as the coefficient of the $x$ term is not a multiple of the coefficient of the $x^2$ term. Equation D: $x^2 - 8 = 1$ can be solved by taking the square root of both sides, but it does not require completing the square.

Based on our analysis, we recommend solving Equation A: $x^2 + 20x = 52$ by completing the square. This method is straightforward and easy to apply, and it will give you the solution to the equation. If you are not comfortable with completing the square, you can also use other methods, such as factoring or the quadratic formula, to solve the equation.

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Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations by manipulating the equation to express it in the form $(x + p)^2 = q$, where $p$ and $q$ are constants.

Q: How do I know if an equation can be solved by completing the square?

A: To determine if an equation can be solved by completing the square, you need to check if the coefficient of the $x$ term is a multiple of the coefficient of the $x^2$ term. If it is, then the equation can be solved by completing the square.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

1. Move the constant term to the right-hand side of the equation.
2. Find the value of $p$ that will allow you to create a perfect square trinomial.
3. Add $p^2$ to both sides of the equation to create a perfect square trinomial.
4. Take the square root of both sides of the equation to solve for $x$.

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

A: No, completing the square is not suitable for all quadratic equations. If the coefficient of the $x$ term is not a multiple of the coefficient of the $x^2$ term, then completing the square is not the best method to use.

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 of the equation.
• Not finding the correct value of $p$ to create a perfect square trinomial.
• Not adding $p^2$ to both sides of the equation.
• Not taking the square root of both sides of the equation.

Q: Can I use completing the square to solve quadratic equations with fractions?

A: Yes, you can use completing the square to solve quadratic equations with fractions. However, you may need to multiply the entire equation by a constant to get rid of the fractions.

Q: Can I use completing the square to solve quadratic equations with negative coefficients?

A: Yes, you can use completing the square to solve quadratic equations with negative coefficients. However, you may need to take the negative square root of both sides of the equation.

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

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

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

Q: Can I use completing the square to solve quadratic equations with complex coefficients?

A: No, completing the square is not suitable for quadratic equations with complex coefficients. In this case, you may need to use other methods, such as the quadratic formula or factoring.

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

A: Some tips for mastering completing the square include:

• Practicing, practicing, practicing!
• Paying attention to the coefficients of the $x$ and $x^2$ terms.
• Using a calculator to check your work.
• Breaking down the problem into smaller steps.

Q: Can I use completing the square to solve quadratic equations with multiple variables?

A: No, completing the square is not suitable for quadratic equations with multiple variables. In this case, you may need to use other methods, such as substitution or elimination.

Q: What are some common mistakes to avoid when using completing the square to solve quadratic equations with multiple variables?

A: Some common mistakes to avoid when using completing the square to solve quadratic equations with multiple variables include:

• Not identifying the correct variables to substitute.
• Not using the correct method to eliminate variables.
• Not checking the solution for extraneous solutions.

In conclusion, completing the square is a powerful algebraic technique that can be used to solve quadratic equations. However, it requires careful attention to the coefficients of the $x$ and $x^2$ terms, as well as a thorough understanding of the steps involved. By following the tips and avoiding common mistakes, you can master completing the square and solve quadratic equations with ease.