Question 439755
The {{{quadratic}}}{{{ formula}}} will calculate the solutions of any {{{quadratic}}}{{{ equation}}}.

    A quadratic equation is an equation that can be written as

        {{{ax^2 + bx + c}}} where a ≠ 0

In other words, a {{{quadratic}}}{{{ equation}}}{{{ must}}}{{{ have}}} a squared term as its highest power. 

The {{{solution}}} of a {{{quadratic}}}{{{ equation}}} is the {{{value}}} of {{{x}}} when you set the equation equal to {{{zero}}},

or when you solve the following general equation: {{{0 = ax^2 + bx + c}}}

    Given a quadratic equation: {{{ax^2 + bx + c}}}

        The quadratic formula below will solve the equation for zero 

    The quadratic formula is : {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}  

Example:

    Quadratic Equation: {{{y = x^2 + 2x + 1}}} where
    {{{a = 1}}}
    {{{b = 2}}}
    {{{c = 1}}}

 Using the quadratic formula to solve this equation just substitute {{{a}}},{{{b}}}, and {{{c}}} into the general formula: 


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

{{{x = (-2 +- sqrt( 2^2-4*1*1 ))/(2*1) }}} 

{{{x = (-2 +- sqrt( 4-4 ))/2 }}}

{{{x = (-2 +- sqrt( 0 ))/2 }}}
 
{{{x = (-2 +-0 )/2 }}}

{{{x = -2 /2 }}}

{{{x = -1 }}}


here is the graph of {{{y = x^2 + 2x + 1}}} and its solution:


{{{ graph(500, 500, -10, 10, -10, 10, x^2 + 2x + 1) }}}


so, as you can see, this quadratic function has only one solution (the parabola is touching x-axis at -1)