Complete The Square To Solve 2 X 2 + 20 X = 10 2x^2 + 20x = 10 2 X 2 + 20 X = 10 .A. X = 2 ± 15 X = 2 \pm \sqrt{15} X = 2 ± 15 ​ B. X = − 2 ± 10 X = -2 \pm \sqrt{10} X = − 2 ± 10 ​ C. X = − 5 ± 30 X = -5 \pm \sqrt{30} X = − 5 ± 30 ​ D. X = 5 ± 30 X = 5 \pm \sqrt{30} X = 5 ± 30 ​

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the most effective methods for solving quadratic equations is the "complete the square" method. This method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will explore the complete the square method and apply it to solve the quadratic equation 2x2+20x=102x^2 + 20x = 10.

What is Complete the Square?

The complete the square method is a technique used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form. This involves adding and subtracting a constant term to create a perfect square trinomial, which can then be easily factored and solved. The method is based on the concept of completing a square, where a perfect square trinomial is created by adding and subtracting a constant term.

Steps to Complete the Square

To complete the square, follow these steps:

  1. Write the equation in standard form: The equation should be written in the standard form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.
  2. Move the constant term to the right-hand side: Move the constant term to the right-hand side of the equation to isolate the quadratic term.
  3. Divide the coefficient of the quadratic term by 2: Divide the coefficient of the quadratic term by 2 and square the result.
  4. Add and subtract the result to the left-hand side: Add and subtract the result to the left-hand side of the equation.
  5. Factor the perfect square trinomial: Factor the perfect square trinomial to solve for the variable.

Solving the Quadratic Equation

Now that we have covered the steps to complete the square, let's apply them to solve the quadratic equation 2x2+20x=102x^2 + 20x = 10.

Step 1: Write the equation in standard form

The equation is already in standard form: 2x2+20x=102x^2 + 20x = 10.

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

Subtract 10 from both sides of the equation to move the constant term to the right-hand side:

2x2+20x−10=02x^2 + 20x - 10 = 0

Step 3: Divide the coefficient of the quadratic term by 2

Divide the coefficient of the quadratic term by 2 and square the result:

22=1\frac{2}{2} = 1

12=11^2 = 1

Step 4: Add and subtract the result to the left-hand side

Add and subtract 1 to the left-hand side of the equation:

2x2+20x+1−1−10=02x^2 + 20x + 1 - 1 - 10 = 0

Step 5: Factor the perfect square trinomial

Factor the perfect square trinomial:

(2x2+20x+1)−11=0(2x^2 + 20x + 1) - 11 = 0

(2x+10)2−11=0(2x + 10)^2 - 11 = 0

(2x+10)2=11(2x + 10)^2 = 11

Step 6: Solve for the variable

Take the square root of both sides of the equation to solve for the variable:

2x+10=±112x + 10 = \pm \sqrt{11}

2x=−10±112x = -10 \pm \sqrt{11}

x=−5±112x = -5 \pm \frac{\sqrt{11}}{2}

Conclusion

In this article, we have explored the complete the square method and applied it to solve the quadratic equation 2x2+20x=102x^2 + 20x = 10. The complete the square method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. By following the steps outlined in this article, you can apply the complete the square method to solve a wide range of quadratic equations.

Final Answer

The final answer is: C\boxed{C}

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Introduction

In our previous article, we explored the complete the square method and applied it to solve the quadratic equation 2x2+20x=102x^2 + 20x = 10. In this article, we will provide a Q&A guide to help you better understand the complete the square method and how to apply it to solve quadratic equations.

Q&A

Q: What is the complete the square method?

A: The complete the square method is a technique used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

  1. Write the equation in standard form: The equation should be written in the standard form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.
  2. Move the constant term to the right-hand side: Move the constant term to the right-hand side of the equation to isolate the quadratic term.
  3. Divide the coefficient of the quadratic term by 2: Divide the coefficient of the quadratic term by 2 and square the result.
  4. Add and subtract the result to the left-hand side: Add and subtract the result to the left-hand side of the equation.
  5. Factor the perfect square trinomial: Factor the perfect square trinomial to solve for the variable.

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

A: An equation can be solved using the complete the square method if it is in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is an expression of the form (x+p)2(x + p)^2, where pp is a constant.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you can use the formula (x+p)2=x2+2px+p2(x + p)^2 = x^2 + 2px + p^2.

Q: What are some common mistakes to avoid when using the complete the square method?

A: Some common mistakes to avoid when using the complete the square method include:

  • Not writing the equation in standard form
  • Not moving the constant term to the right-hand side
  • Not dividing the coefficient of the quadratic term by 2
  • Not adding and subtracting the result to the left-hand side
  • Not factoring the perfect square trinomial

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

A: No, the complete the square method can only be used to solve quadratic equations that are in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

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

A: The complete the square method has many real-world applications, including:

  • Solving quadratic equations in physics and engineering
  • Modeling population growth and decay
  • Analyzing data and making predictions
  • Solving optimization problems

Conclusion

In this article, we have provided a Q&A guide to help you better understand the complete the square method and how to apply it to solve quadratic equations. By following the steps outlined in this article, you can apply the complete the square method to solve a wide range of quadratic equations.

Final Tips

  • Make sure to write the equation in standard form before attempting to solve it using the complete the square method.
  • Be careful when dividing the coefficient of the quadratic term by 2 and squaring the result.
  • Make sure to add and subtract the result to the left-hand side of the equation.
  • Factor the perfect square trinomial carefully to avoid making mistakes.

Final Answer

The final answer is: C\boxed{C}