Complete The Square For $5x^2 - 30x = 5$.A. $(5x - 3)^2 = 14$B. \$(x - 3)^2 = 10$[/tex\]C. $(x + 3)^2 = 10$D. $(5x - 3)^2 = 8$

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. In this article, we will learn how to complete the square for the quadratic equation $5x^2 - 30x = 5$.

Step 1: Move the Constant Term to the Right Side

The first step in completing the square is to move the constant term to the right side of the equation. This gives us:

5x2βˆ’30xβˆ’5=05x^2 - 30x - 5 = 0

Step 2: Factor Out the Coefficient of x2x^2

Next, we factor out the coefficient of x2x^2, which is 5. This gives us:

5(x2βˆ’6x)βˆ’5=05(x^2 - 6x) - 5 = 0

Step 3: Add and Subtract the Square of Half the Coefficient of xx

Now, we need to add and subtract the square of half the coefficient of xx inside the parentheses. The coefficient of xx is -6, so half of it is -3. The square of -3 is 9. We add and subtract 9 inside the parentheses:

5(x2βˆ’6x+9βˆ’9)βˆ’5=05(x^2 - 6x + 9 - 9) - 5 = 0

Step 4: Simplify the Equation

Next, we simplify the equation by combining like terms:

5(x2βˆ’6x+9)βˆ’5βˆ’45=05(x^2 - 6x + 9) - 5 - 45 = 0

Step 5: Factor the Perfect Square

Now, we can factor the perfect square:

5(xβˆ’3)2βˆ’50=05(x - 3)^2 - 50 = 0

Step 6: Add 50 to Both Sides

Next, we add 50 to both sides of the equation to isolate the perfect square:

5(xβˆ’3)2=505(x - 3)^2 = 50

Step 7: Divide Both Sides by 5

Finally, we divide both sides of the equation by 5 to solve for the perfect square:

(xβˆ’3)2=10(x - 3)^2 = 10

Conclusion

In this article, we learned how to complete the square for the quadratic equation $5x^2 - 30x = 5$. We moved the constant term to the right side, factored out the coefficient of x2x^2, added and subtracted the square of half the coefficient of xx, simplified the equation, factored the perfect square, added 50 to both sides, and finally divided both sides by 5 to solve for the perfect square. The final answer is:

(xβˆ’3)2=10(x - 3)^2 = 10

This is option B.

Answer Key

A. $(5x - 3)^2 = 14$ B. $(x - 3)^2 = 10$ C. $(x + 3)^2 = 10$ D. $(5x - 3)^2 = 8$

The correct answer is B. $(x - 3)^2 = 10$

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. 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 technique used to solve quadratic equations by manipulating the equation to express it in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: Why do we need to complete the square?

A: We need to complete the square to solve quadratic equations that cannot be factored easily. By completing the square, we can express the equation in a perfect square form, which can be easily solved.

Q: How do I know when to complete the square?

A: You should complete the square when you have a quadratic equation that cannot be factored easily. This is usually the case when the equation has a coefficient of 1 on the x2x^2 term.

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 side of the equation.
  2. Factor out the coefficient of x2x^2.
  3. Add and subtract the square of half the coefficient of xx inside the parentheses.
  4. Simplify the equation.
  5. Factor the perfect square.
  6. Add the constant term to both sides of the equation.
  7. Divide both sides of the equation by the coefficient of the perfect square.

Q: What is the difference between completing the square and factoring?

A: Factoring involves expressing a quadratic equation as a product of two binomials. Completing the square involves expressing a quadratic equation in a perfect square form.

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

A: No, completing the square is not suitable for all types of quadratic equations. It is best used for quadratic equations that cannot be factored easily.

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 side of the equation.
  • Not factoring out the coefficient of x2x^2.
  • Not adding and subtracting the square of half the coefficient of xx inside the parentheses.
  • Not simplifying the equation.
  • Not factoring the perfect square.

Q: How do I check my work when completing the square?

A: To check your work when completing the square, you can plug the solution back into the original equation to see if it is true.

Conclusion

In this article, we answered some frequently asked questions about completing the square. We discussed the steps to complete the square, the difference between completing the square and factoring, and some common mistakes to avoid. We also provided some tips for checking your work when completing the square.

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