Consider The Equation:$0 = X^2 - 10x + 10$1) Rewrite The Equation By Completing The Square.Your Equation Should Look Like ( X + A ) 2 = B (x + A)^2 = B ( X + A ) 2 = B Or ( X − C ) 2 = D (x - C)^2 = D ( X − C ) 2 = D .2) What Are The Solutions To The Equation?

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the methods used to solve quadratic equations is completing the square. This method involves rewriting the equation in a specific form, which allows us to easily identify the solutions. In this article, we will explore how to complete the square and solve the quadratic equation $0 = x^2 - 10x + 10$.

Completing the Square

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Completing the square is a technique used to rewrite a quadratic equation in the form $(x + a)^2 = b$ or $(x - c)^2 = d$. This form is useful because it allows us to easily identify the solutions to the equation. To complete the square, we need to follow these steps:

1. Write the equation in the standard form: The equation should be in the form $ax^2 + bx + c = 0$.
2. Move the constant term to the right-hand side: This will give us the equation $ax^2 + bx = -c$.
3. Divide both sides by $a$: This will give us the equation $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
4. Add and subtract the square of half the coefficient of $x$: This will give us the equation $x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a}$.
5. Write the left-hand side as a perfect square: This will give us the equation $\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a}$.

Applying Completing the Square to the Given Equation

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Now that we have the steps to complete the square, let's apply them to the given equation $0 = x^2 - 10x + 10$.

Step 1: Write the equation in the standard form

The equation is already in the standard form $ax^2 + bx + c = 0$, where $a = 1$, $b = -10$, and $c = 10$.

Step 2: Move the constant term to the right-hand side

Moving the constant term to the right-hand side gives us the equation $x^2 - 10x = -10$.

Step 3: Divide both sides by $a$

Dividing both sides by $a = 1$ gives us the equation $x^2 - 10x = -10$.

Step 4: Add and subtract the square of half the coefficient of $x$

Adding and subtracting the square of half the coefficient of $x$ gives us the equation $x^2 - 10x + 25 - 25 = -10$.

Step 5: Write the left-hand side as a perfect square

Writing the left-hand side as a perfect square gives us the equation $(x - 5)^2 - 25 = -10$.

Simplifying the Equation

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Now that we have completed the square, let's simplify the equation.

$(x - 5)^2 - 25 = -10$

Adding 25 to both sides gives us:

$(x - 5)^2 = 15$

Solving the Equation

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Now that we have the equation in the form $(x - c)^2 = d$, we can easily identify the solutions.

$(x - 5)^2 = 15$

Taking the square root of both sides gives us:

$x - 5 = \pm \sqrt{15}$

Adding 5 to both sides gives us:

$x = 5 \pm \sqrt{15}$

Conclusion

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In this article, we have learned how to complete the square and solve the quadratic equation $0 = x^2 - 10x + 10$. We have followed the steps to complete the square and simplified the equation to find the solutions. The solutions to the equation are $x = 5 \pm \sqrt{15}$. This method is useful for solving quadratic equations and is an important concept in mathematics.

Final Answer

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The final answer is $\boxed{5 \pm \sqrt{15}}$.

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Introduction

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In our previous article, we explored how to complete the square and solve quadratic equations. In this article, we will answer some frequently asked questions about completing the square and solving quadratic equations.

Q: What is completing the square?

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A: Completing the square is a technique used to rewrite a quadratic equation in the form $(x + a)^2 = b$ or $(x - c)^2 = d$. This form is useful because it allows us to easily identify the solutions to the equation.

Q: How do I complete the square?

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

1. Write the equation in the standard form: The equation should be in the form $ax^2 + bx + c = 0$.
2. Move the constant term to the right-hand side: This will give us the equation $ax^2 + bx = -c$.
3. Divide both sides by $a$: This will give us the equation $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
4. Add and subtract the square of half the coefficient of $x$: This will give us the equation $x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a}$.
5. Write the left-hand side as a perfect square: This will give us the equation $\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a}$.

Q: What are the advantages of completing the square?

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A: The advantages of completing the square are:

• It allows us to easily identify the solutions to the equation.
• It is a useful technique for solving quadratic equations.
• It can be used to solve equations that are not easily solvable using other methods.

Q: What are the disadvantages of completing the square?

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A: The disadvantages of completing the square are:

• It can be a time-consuming process.
• It requires a good understanding of algebraic manipulations.
• It may not be the most efficient method for solving certain types of equations.

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

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A: No, completing the square cannot be used to solve all types of quadratic equations. It is most useful for solving equations that can be written in the form $(x + a)^2 = b$ or $(x - c)^2 = d$.

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

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A: To determine if an equation can be solved using completing the square, you need to check if the equation can be written in the form $(x + a)^2 = b$ or $(x - c)^2 = d$. If it can, then completing the square can be used to solve the equation.

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

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

• Not following the steps correctly.
• Not adding and subtracting the square of half the coefficient of $x$.
• Not writing the left-hand side as a perfect square.

Conclusion

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In this article, we have answered some frequently asked questions about completing the square and solving quadratic equations. We have discussed the advantages and disadvantages of completing the square, and provided some tips for avoiding common mistakes. Completing the square is a useful technique for solving quadratic equations, and with practice, it can become a valuable tool in your mathematical toolkit.

Final Answer

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The final answer is $\boxed{5 \pm \sqrt{15}}$.