Activity: To find the solutionof quadratic equations by completing square method
Answers
Answer:
e have our quadratic equation as;
ax2 + bx + c = 0
The roots of the quadratic equation can also be evaluated by using the quadratic formula. For simplification, let us take a=1. The equation hence becomes
x2 + bx + c = 0
If we wanted to represent a quadratic equation using geometry, one way would be representing the terms of the expression in the L.H.S. of the equation by using geometric figures such as squares, rectangles etc. If we take a square with the side equal to x units, its area would be equal to x2 square units. This area will hence represent the first term of the expression. Similarly, a rectangle with its two sides as x units and b units will have the area equal to bx square units. And let us take a square with area equal to c square units to represent the last term of the expression. In the figure below, we have the geometrical equivalent of the expression x2, bx and c.
Completing the Square method
Geometrical equivalent of x2,bx and c
e have our quadratic equation as;
ax2 + bx + c = 0
The roots of the quadratic equation can also be evaluated by using the quadratic formula. For simplification, let us take a=1. The equation hence becomes
x2 + bx + c = 0
If we wanted to represent a quadratic equation using geometry, one way would be representing the terms of the expression in the L.H.S. of the equation by using geometric figures such as squares, rectangles etc. If we take a square with the side equal to x units, its area would be equal to x2 square units. This area will hence represent the first term of the expression. Similarly, a rectangle with its two sides as x units and b units will have the area equal to bx square units. And let us take a square with area equal to c square units to represent the last term of the expression. In the figure below, we have the geometrical equivalent of the expression x2, bx and c.
Completing the Square method
Geometrical equivalent of x2,bx and c
This method is known as completing the square. Let us complete some squares. If we break the rectangle representing bx into two equal parts cutting vertically, we will have two figures with an area of each equal to b/2 x square units. The figures are arranged accordingly in the second figure, below. Now, we have
x2 + bx + c = 0
Completing the square
Rearranging the figures
But our square is not complete yet. To complete the square, one square of side b/2 units is needed. This final part of the main square can be taken from the square with the area c square units. Cutting it out and putting it at the place, it results in below figure.
Completing square method
Completing the Square
The square is finally complete. The area of the square is equal to
(x+b/2)2 square units
The remaining area is equal to
(c-b2/4) square units
All this time, we were rearranging the same figures that we had initially. It would hence be correct to say that
x2 + bx + c = (x+b/2)2 + (c-b2/4)
This method is known as completing the square method. We have achieved it geometrically. We know that x2 + bx + c = 0. So,
(x+b/2)2 + (c-b2/4) = 0
⇒ (x+b/2)2 = -(c-b2/4)
All the terms in the R.H.S. of the above equation are known. That’s why it is very easy to determine the roots. Let us look at some examples for better understanding.
This method is known as completing the square. Let us complete some squares. If we break the rectangle representing bx into two equal parts cutting vertically, we will have two figures with an area of each equal to b/2 x square units. The figures are arranged accordingly in the second figure, below. Now, we have
x2 + bx + c = 0
Completing the square
Rearranging the figures
But our square is not complete yet. To complete the square, one square of side b/2 units is needed. This final part of the main square can be taken from the square with the area c square units. Cutting it out and putting it at the place, it results in below figure.
Completing square method
Completing the Square
The square is finally complete. The area of the square is equal to
(x+b/2)2 square units
The remaining area is equal to
(c-b2/4) square units
All this time, we were rearranging the same figures that we had initially. It would hence be correct to say that
x2 + bx + c = (x+b/2)2 + (c-b2/4)
This method is known as completing the square method. We have achieved it geometrically. We know that x2 + bx + c = 0. So,
(x+b/2)2 + (c-b2/4) = 0
⇒ (x+b/2)2 = -(c-b2/4)
All the terms in the R.H.S. of the above equation are known. That’s why it is very easy to determine the roots. Let us look at some examples for better understanding.
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