Question

state the general formula of quadratic equation.obtain the solution of this equation .using the method of completing a square.also prove alpha +beta=-b÷a

Answer 1

Answer:

Step-by-step explanation:

General Formula of a Quadratic Equation:- ax² + bx + c = 0

Solution using Completing The Square Method:

Step 1: Write the equation

⇒ ax² + bx + c = 0

Step 2: Divide by the coefficient of x²

⇒ x² + ( b/a ) x + c/a = 0

Step 3: Transpose c/a to the RHS.

⇒ x² + ( b/a ) x = - c/a

Step 4: Divide the coefficient of x by 2 and square the result. Then add it on both sides.

⇒ Coefficient of x = b/a

On dividing by 2, we get the result as b/a ÷ 2 = b/2a

Now ( b/ 2a )² must be added on both sides.

⇒ x² + b/a + ( b/ 2a )² = ( b/ 2a )² - c/a

Now LHS is of the form of a² + 2ab + b². Hence we can write the LHS as,

⇒ ( x + b/2a )² = ( b/ 2a )² - c/a

⇒ ( x + b/2a )² = b²/ 4a² - c/a

Taking LCM on RHS we get,

⇒ ( x + b/2a )² = b² -4ac / 4a²

Step 5: Take Square root on both sides

⇒ ( x + b/2a ) = √( b² -4ac / 4a² )

The denominator 4a² will come outside the root as ±2a.

⇒ ( x + b/2a ) = ± √( b² -4ac ) / 2a

Step 6: Transpose b/2a from LHS.

⇒ x = - b/2a ± √( b² -4ac ) / 2a

⇒ x = [ -b ± √( b² -4ac ) ] / 2a

Hence this is the solution of the quadratic equation.

Now we have two roots: [ -b + √( b² -4ac ) / 2a and [ -b - √( b² -4ac ) / 2a ]

Now Sum of roots will give us:

⇒  [ -b + √( b² -4ac ) / 2a ] + [ -b - √( b² -4ac ) / 2a ]

+ √( b² -4ac )  and -√( b² -4ac ) cancel each other and hence we get,

⇒ -2b / 2a

⇒ -b / a

Hence Sum of Roots is -b / a

Thanks !!