Question 999340

the discriminant of {{{sqrt((-12)^2 - 4 * 9 * 3)=sqrt(144 - 108)=sqrt(36)}}}

so, the discriminant is {{{36}}} and {{{36>0}}};  that means quadratic function has {{{two}}} {{{real}}} solutions 

now, we can find what is the equation:

recall: the discriminant is  {{{b^2 - 4 * a* c}}}

when you compare it to given above,  {{{(-12)^2 - 4 * 9 * 3)}}}, you see that {{{b= -12}}}, {{{a=9}}}, and {{{c=3}}}

so, the possible quadratic function is {{{y=9x^2-12x+3}}} and it has {{{two}}} {{{real}}} solutions

let's find them:
make {{{y=0}}}
{{{0=9x^2-12x+3}}}...both sides divide by {{{3}}}

{{{0=3x^2-4x+1}}}....factor

{{{0=3x^2-3x-x+1}}}...group

{{{0=(3x^2-3x)-(x-1)}}}

{{{0=3x(x-1)-(x-1)}}}

{{{0=(3x-1)(x-1)}}}

solutions:

if {{{0=(3x-1)}}}=>{{{3x=1}}}=>{{{highlight(x=1/3)}}}
{{{0=(x-1)}}}=>{{{highlight(x=1)}}}

see it on the graph:
{{{ graph( 600, 600, -5, 5, -10, 10, 9x^2-12x+3) }}}