Question 1126098
{{{x^2 - 2mx +8m -15 =0}}}


{{{x^2 - 2mx +8m -15 =0}}}..-> compare to {{{ax^2+bx+c=0}}} and you see that 

{{{a=1}}}, {{{b=2m}}}, and {{{c=8m-15}}}

use discriminant to find for what values of m has the equation will have certain number of solutions

discriminant is  {{{b^-4ac}}} (portion of the quadratic formula {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 


a) one root?

if {{{b^2-4ac=0}}} there is exactly one root

{{{b^2-4ac=0}}}....plug in {{{a=1}}}, {{{b=2m}}}, and {{{c=8m-15}}}

{{{(2m)^2−4*1(8m-15)=0}}}
{{{4m^2-32m+60=0}}}...simpliy
{{{m^2-8m+15=0}}}.
{{{(m - 5) (m - 3) = 0}}}

{{{m = 5}}}  or {{{m = 3}}}



check:{{{m = 5}}}

{{{x^2 - 2*5x +8*5 -15 =0}}}
{{{x^2 - 10x +40 -15 =0}}}
{{{x^2 - 10x +25 =0}}}

{{{ graph( 600, 600, -10, 10, -10, 10,x^2 - 10x +25) }}}

check:{{{m = 3}}}

{{{x^2 - 2*3x +8*3 -15 =0}}}
{{{x^2 - 6x +24 -15 =0}}}
{{{x^2 - 6x +9 =0}}}

{{{ graph( 600, 600, -10, 10, -10, 10,x^2 - 6x +9) }}}




b) no roots?

if {{{b^2-4ac<0}}} there are no Real roots

{{{b^2-4ac<0}}}....plug in {{{a=1}}}, {{{b=2m}}}, and {{{c=8m-15}}}

{{{(2m)^2-4*1(8m-15)<0}}}
{{{4m^2-32m+60<0}}}
{{{(m - 5) (m - 3) < 0}}}
{{{m < 5}}} or {{{m > 3}}}



check: if {{{m < 5}}}, let say {{{m = 4}}}

{{{x^2 - 2*4x +8*4 -15 =0}}}
{{{x^2 - 8x +32 -15 =0}}}
{{{x^2 - 8x +17 =0}}}

{{{ graph( 600, 600, -10, 10, -10, 10,x^2 - 8x +17) }}}


check:if {{{m > 3}}}, let say {{{m =3.5}}}

{{{x^2 - 2*3.5x +8*3.5 -15 =0}}}
{{{x^2 - 7x +28 -15 =0}}}
{{{x^2 - 7x +13 =0}}}

{{{ graph( 600, 600, -10, 10, -10, 10,x^2 - 7x +13 ) }}}



c) two roots? 

if {{{b^2-4ac>0}}} there are two roots

{{{b^2-4ac<0}}}....plug in {{{a=1}}}, {{{b=2m}}}, and {{{c=8m-15}}}

{{{(2m)^2-4*1(8m-15)>0}}}
{{{4m^2-32m+60>0}}}
{{{(m - 5) (m - 3) >0}}}

{{{m > 5}}} or {{{m < 3}}}



check: if {{{m > 5}}}, let say {{{m = 6}}}

{{{x^2 - 2*6x +8*6 -15 =0}}}
{{{x^2 - 12x +48 -15 =0}}}
{{{x^2 - 12x +33 =0}}}

{{{ graph( 600, 600, -10, 10, -10, 10,x^2 - 12x +33) }}}

check:if {{{m < 3}}}, let say {{{m = 2}}}


{{{x^2 - 2*2x +8*2 -15 =0}}}
{{{x^2 - 4x +16 -15 =0}}}
{{{x^2 - 4x +1 =0}}}

{{{ graph( 600, 600, -10, 10, -10, 10,x^2 - 4x +1) }}}