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## Answers

ax2 + bx = –c

Divide through by whatever is multiplied on the squared term.

x2 + (b / a) x = - c / a

and simplify

x2 + (b / a) x = - c / a

Add (b / 2a)2 to both sides

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

to complete the square

(x + b / 2a )2 = - c / a + (b / 2a)2

Group the two terms on the right side of the equation

(x + b / 2a)2 = (b2 - 4 a c) / (4 a2 )

Solve by taking the square root

x + b / 2a = ± √{ (b2 - 4 a c ) / (4 a2) }

Solve for x to obtain two solutions

x = - b / 2a ± √{ (b2 - 4 a c ) / (4 a2) }

The term √{ (b2 - 4 a c ) / (4 a2) }may be simplified as follows

√{ (b2 - 4 a c ) / (4 a2) } = √(b2 - 4 a c) / (2 |a|)

Since 2 | a | = 2 a for a > 0 and 2 | a | = -2 a for a < 0, the two solutions to the quadratic equation may be written

x = (-b + √( b2 - 4 a c)) / (2 a)

x = (-b - √ ( b2 - 4 a c)) / (2 a)