Quadratic Formula first created 8/11/06 - last modified 12/11/06 Page Author: Ty Harness

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The general quadratic equation is often written in the form:
$ax^2 + bx + c = 0 $ .......(1)

To find the 2 values of x we can manipulate the formula to obtain what is called the quadratic formula (equation 16) which all students memorize but it's well worth looking at the algebraic manipulation because you never know when these techniques will crop up.

$x^2 + b/a x + c/a = 0 $ ......(2)

$x^2 + b/a x = - c/a$ ......(3)

If we remember $(alpha + beta)^2 $ square can be multiplied out and we can see a comparison between equation 3 and 4 and $beta$ can be defined.

$(alpha + beta)^2 = (alpha + beta)*(alpha + beta) = alpha^2 + 2 alpha beta + beta^2$ ......(4)

$x = alpha$ ......(5)

$alpha^2 + b/a alpha = alpha^2 + 2 alpha beta$ ......(6)

$b/a alpha = 2 alpha beta$ ......(7)

$b/(2a) = beta$ .....(8)

Now to complete the square similar to equation 4 we need to add $beta^2$ to the LHS but what we do to the LHS we must do the same to the RHS to balance the equation.

$x^2 + b/a x + (b/(2a))^2 = (b/(2a))^2 - c/a$ .....(9)

$(x + b/(2a))^2 = (b/(2a))^2 - c/a$ .....(10)

$(x + b/(2a))^2 = b^2/(4a^2) - c/a$ .....(11)

$(x + b/(2a))^2 = (a b^2 - 4a^2c)/(4a^3) $ .....(12)

$(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2) $ .....(13)

Remember taking the square root yeilds 2 results + and -.

$x + b/(2a) = +- sqrt( (b^2 - 4ac)) / sqrt(4a^2) $ .....(14)

$x = - b/(2a) +- sqrt( (b^2 - 4ac)) / (2a) $ .....(15)

$x = (- b+- sqrt( (b^2 - 4ac)) )/ (2a) $ .....(16)

Equation 16 is known as the quadratic formula.

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