What Is The Solution To The Quadratic Equation?

Given
ax 2 + bx + c = 0


1. Divide all terms by a -->
x 2 + (b / a) x + c / a = 0


2. Subtract c / a from both sides -->
x 2 + (b / a) x + c / a - c / a = - c / a


3. Simplify -->
x 2 + (b / a) x = - c / a


4. Add (b / 2a) 2 to both sides -->
x 2 + (b / a) x + (b / 2a) 2 = - c / a + (b / 2a) 2


5. Complete the square -->
[ x + (b / 2a) ] 2 = - c / a + (b / 2a) 2


6. Group the two terms on the right side of the equation -->
[ x + (b / 2a) ] 2 = [ b 2 - 4a c ] / ( 4a2 )


7. Solve by taking the square root -->
x + (b / 2a) = ± sqrt { [ b 2 - 4a c ] / ( 4a2 ) }


8. Solve for x to obtain two solutions -->
x = - b / 2a ± sqrt { [ b 2 - 4a c ] / ( 4a2 ) }


--> The term sqrt { [ b 2 - 4a c ] / ( 4a2 ) } may be written
sqrt { [ b 2 - 4a c ] / ( 4a2 ) } = sqrt(b 2 - 4a c) / 2 | a |


--> Since 2 | a | = 2a when a > 0 and 2 | a | = -2a when a < 0, -->
solution 1 -- > x = [ -b + sqrt( b 2 - 4a c ) ] / 2 a -->

Solution 2 --> x = [ -b - sqrt( b 2 - 4a c ) ] / 2 a

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In mathematics, the quadratic equation is a polynomial equation of the second degree. The general form is ax2 + bx + c = 0
The letters a, b, and c are called as coefficients: The quadratic coefficient a is the coefficient of x2, linear coefficient b is the coefficient of x, and c is the called constant coefficient and also called the free term.
For more information, see the link below:
en.wikipedia.org

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