Question 587200
Look at {{{b^2 - 4*a*c }}} where the
equation has the form {{{ a*x^2 + b*x +c = 0 }}}
(1)
{{{ b^2 - 4*a*c = (-10)^2 - 4*25*1 }}}
{{{ b^2 - 4*a*c = 100 - 100 }}}
{{{ b^2 - 4*a*c = 0 }}}
This means there is 1 real solution
It is called a double root
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(2)
{{{ b^2 - 4*a*c = 1^2 - 4*1*2 }}}
{{{ b^2 - 4*a*c = 1 - 8 }}}
{{{ b^2 - 4*a*c = -7 }}}
This is minus, so there are 2
imaginary roots ( look like {{{a+b*i}}} and {{{a-b*i}}} )
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(3)
{{{ b^2 - 4*a*c = (-1)^2 - 4*5*0 }}}
{{{ b^2 - 4*a*c = 1 }}}
This is plus, so there are 2 real roots
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Here's plots of the 3 equations.
Touching the x-axis: 1 solution
Not touching at all: 2 imaginary solutions
Touching twice: 2 real solutions
{{{ graph( 400, 400, -2, 2, -2, 5, 25x^2-10x+1,x^2+x+2,5x^2-x ) }}}