Describe the method of 'Completing the squares' with an example and explain idea behind it.
Answers
METHOD OF COMPLETING THE SQUARE :
Step 1 - Write the given equation in standard form, ax²+bx+c = 0, a≠0.
Step 2 - If the coefficient of x² is 1, go to step 3. If not, divide both sides of the equation by the coefficient of x²
Step 3 - Shift the constant term (c/a) on RHS.
Step 4- Find half the coefficient of x and square it. Add this number to both sides of the equation.
Step 5 - Write LHS in the form a perfect square and simplify the RHS.
Step 6 - Take the square root on both sides.
Step 7 : Find the values of x by shifting the constant term(b/2a) on RHS from LHS.
EXAMPLE:
x² + 3x - 9 = 0
=> x² + 3x = 9
=> x² + 3x + (3/2)² = (3/2)² + 9
=> x² + 2×(3/2)× x + (3/2)²= 9 + (3/2)²
[a² +2ab + b² = (a+b)²]
=> (x + 3/2)² = 9/4 + 9
=> (x + 3/2)² = (9+36)/4
=> (x + 3/2)² = 45/4
=> (x + 3/2) = √(45/4)
=> (x + 3/2) = √(9×5/4)
=> (x + 3/2) = ±3√5/2
=> x = -3/2 ± 3√5/2
=>x = (-3 ± 3√5)/2
Hence, roots of the quadratic equation are (-3 - 3√5)/2 and (-3 + 3√5)/2.
HOPE THIS ANSWER WILL HELP YOU...
Algorithm :
Let the quadratic equation be
ax² + bx + c = 0
step 1 : Divide each side by ' a '
step 2 : Rearrange the equation so
that constant term c/a is on the right
side ( RHS )
step 3 : Add [ 1/2(b/a )² ] to both sides
to make LHS , a perfect square .
step 4 : Write the LHS As a square
and simplify the RHS .
step 5 : Solve it .
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example :
Given : 5x² - 6x - 2 = 0
Now we follow the algorithm
step 1 : x² - (6/5)x - 2/5 = 0
( dividing both sides by 5 )
step 2 : x² - (6/5 )x = 2/5
step 3 : x²- (6/5)x + (3/5)² = 2/5 +( 3/5)²
{ [ 1/2[ (b/a ) ]² =( 3/5 )² }
[ adding ( 3/5 )² to both sides ]
Step 4 : ( x - 3/5 )² = 2/5 + 9/25
step 5 : ( x - 3/5 )² = 19/25
=> x - 3/5 = ± √(19/25)
x = 3/5 ± √19/5
Therefore ,
x = ( 3 + √19 )/5 or x = ( 3 - √19 )/5
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# The idea behind this method is to
adjust the left side of the quadratic
equation so that it becomes a perfect
square.
The square of a first degree polynomial.
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I hope this helps you.
: )