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Completing the Square:

It is a technique used to transform a standard quadratic equation, such that one side of the equation turns into a perfect square.


Following are the steps for Completing the Square of the quadratic equation in the form of ax2+bx+c=0.
 

  • Rewrite the equation in the form of ax+bx = -c, and divide both sides by a.
  • Add (b/2a)2 to both sides of the equation.
  • Simplify the left side as a perfect square, and combine the right-side terms.
  • Then, take the square root on both sides of the equation.
  • Finally, solve for x.

 

Completing the Square Sample Questions:

Question 1: What number should be added to x2(x/2) to complete the square.


Answer:  1/16

Explanation: To complete the square, we need to add (b/2a)2 to the equation.

 

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Question 2: What number should be added to x2 - (2x/3) to complete the square.


Answer: 1/9

Explanation: To complete the square, we need to add (b/2a)2 to the equation.

 

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Question 3: Complete the square of the following equation: x+ 3x+ 7=0


Answer: (x + 3/2)2  = -19/4

Explanation: To complete the square, we need to write the equation in the form of x+ bx/a = -c/a , and then add (b/2a)2 to both sides of the equation.

 

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Question 4: Complete the square of the following equation: x2- 5x - 2 = 0


Answer: (x - 5/2)2  = 33/4

Explanation: To complete the square, we need to write the equation in the form of x+ bx/a = -c/a , and then add (b/2a)2 to both sides of the equation.

 

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Question 5: What number should be added to 3x- 5x to complete the square.


Answer: 25/36

Explanation: To complete the square, we need to divide the equation by a, and then add (b/2a)2 to the equation.

 

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Question 6: What number should be added to 5x2 + 4x to complete the square?


Answer: 4/25

Explanation: To complete the square, we need to divide the equation by a, and then add (b/2a)2 to the equation.

 

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Question 7: Solve x2 -6x - 12 = 0 by completing the square.


Answer: x = 3±√21

Explanation: To complete the square, we need to add (b/2a)2 to the equation.

 

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Question 8: Solve y2 + 8y - 7 = 0 by completing the square.


Answer: y = 4±√23

Explanation: To complete the square, we need to add (b/2a)2 to the equation.

 

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Question 9: Complete the square: 3y2 + 5y + 4 = 0


Answer: (y + 5/6)2  = -23/36

Explanation: To complete the square, we need to divide the equation by a, and then add (b/2a)2 to the equation.

 

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Question 10: Complete the square: 3y2 + 9y + 10 = 0


Answer: y = -4±√23

Explanation: Plug the value of height and total surface area in the total surface area of cyclinder formula.

 

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Completing The Square Frequently Asked Questions:

Question 1: How to solve by completing the square?

The method of completing the square is to change the form of a quadratic equation so that the left side is a perfect square.

To solve the quadratic equation, ax2 + bx + c = 0 , we have to convert ax2 + bx + c = 0 to a(x+d)2 + e = 0

On solving, d = (b/2a)

and, e = c - (b2/4a)

 

Question 2: What is completing the square ?
A method for converting a quadratic polynomial or equation into a perfect square with the addition of a constant is known as the completing square.

Using the completing the square formula or approach, a quadratic equation in variable x: ax2 + bx +c =0

where a, b, and c are any real integers except a 0, may be transformed into a perfect square using one additional constant.

 

Question 3: How to solve quadratic equations by completing the square?
To solve the quadratic equation, ax2 + bx +c =0 , we have to convert ax2 + bx + c = 0 to a(x+d)2 + e = 0

On solving, d = (b/2a)

and, e = c - (b2/4a)

 

Question 4: Solve x2 = 12x – 15 by completing the square. Which is the solution set of the equation?


answer 4

 

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