7.3 The Quadratic Formula

  • The Quadratic Formula • Character of the Roots of a Quadratic Equation

We now use the method of completing the square to derive a general formula that may be used for the solution of any quadratic equation.

Consider the general quadratic equation:

ax2 + bx + c = 0(a ≠ 0)

When we divide through by a, we obtain

x2 + bax + ca = 0

Subtracting c/a from each side, we have

x2 + bax =  − ca

Half of b/a is b/2a, which squared is b2 / 4a2 .  Adding b2 / 4a2 to each side gives us

x2 + bax + b24a2 =  − ca + b24a2

Writing the left side as a perfect square and combining fractions on the right side,

(x + b2a)2 = b2 − 4ac4a2

Equating x + b2a to the positive and negative square root of the right side,

x + b2a =  ± b2 − 4ac2a

When ...

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