Completing a square and quadratic equation solution

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I Method

Methods of completing a square and to derive the formula for the solution of the quadratic equation.

Step 1:

Divide by 'a' throughout.

II Method ( By Sridhar's Method)

Methods of completing a square and to derive the formula for the solution of the quadratic equation.

Step 1:

Multiplying by 4 times the coefficient of x² on both sides i.e., 4a.

Step 2:

4a²x² + 4abx = -4ac [Transposing 4ac to RHS]

Step 3:

(2ax)² + 2(2ax)b = -4ac [Writing 4a²x² = (2ax)²,

4abx = 2(2ax)b]

Step 4:

(2ax)² + 2(2ax)b +b²= b² - 4ac

[Adding b² to both sides (square of the coefficient of x)]

Step 5:

(2ax + b)² = b² - 4ac [a² + 2ab + b² = (a+b)²]

Note:

1.

i) Upto step 6 in the I Method and upto Step 7 in the II Method, gives the process of completing the square.

ii) The quadratic equation has two roots.

(i.e., there are two values for x)

is b² - 4ac. This is usually denoted by D or D. D determines the nature of the roots.