Question 953081
the general form is:
{{{ y = a*x^2 + b*x + c }}}
( 1,0 )
{{{ 0 = a*1^2 + b*1 + c }}}
(1) {{{ a + b + c = 0 }}}
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( 2,5 )
{{{ 5 = a*2^2 + b*2 + c }}}
(2) {{{ 4a + 2b + c = 5 }}}
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( 4,21 )
{{{ 21 = a*4^2 + b*4 + c }}}
(3) {{{ 16a + 4b + c = 21 }}}
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Subtract (1) from (2)
(2) {{{ 4a + 2b + c = 5 }}}
(1) {{{ -a - b - c = 0 }}}
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{{{ 3a + b = 5 }}}
Multiply both sides by {{{ 4 }}}
and subtract this result from (3)
{{{ 16a + 4b + c = 21 }}}
{{{ -12a - 4b = -20 }}}
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{{{ 4a + c = 1 }}}
{{{ c = -4a + 1 }}}
Plug this into (1)
(1) {{{ a + b + -4a + 1 = 0 }}}
also, from above:
{{{ 3a + b = 5 }}}
{{{ b = -3a + 5 }}}
Plug this into (1)
(1) {{{ a + -3a + 5 + -4a + 1 = 0 }}}
(1) {{{ -6a  = -6 }}}
(1) {{{ a = 1 }}}
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{{{ c = -4a + 1 }}}
{{{ c = -4*1 + 1 }}}
{{{ c = -3 }}}
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{{{ b = -3a + 5 }}}
{{{ b = -3*1 + 5 }}}
{{{ b = 2 }}}
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Plug these results into:
{{{ y = a*x^2 + b*x + c }}}
{{{ y = x^2 + 2x - 3 }}}
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check:
( 4,21 )
{{{ 21 = 4^2 + 2*4 - 3 }}}
{{{ 21 = 16 + 8 - 3 }}}
{{{ 21 = 21 }}}
You can check the other 2 points