Question 79946
{{{y[1] = a(x-3)^2+c}}}
{{{y[2] = (2x-3)(2x+b)}}}
Expand both equations:
{{{y[1] = a(x^2-6x+9)+c}}}
{{{y[1] = ax^2-6ax+(9a+c)}}}
{{{y[2] = 4x^2+(2b-6)x-3b}}}
Since these two equations represent the same quadratic function, the corresponding terms should be equal, so comparing term-for-term,...
First term:
{{{ax^2 = 4x^2}}}
{{{a = 4}}}
Second term:
{{{-6ax = (2b-6)x}}}
{{{-6a = 2b-6}}} Substitute a = 4.
{{{-6(4) = 2b-6}}}
{{{-24 = 2b-6}}}
{{{2b = -18}}}
{{{b = -9}}}
Third term:
{{{9a+c = -3b}}} Substitute a = 4 and b = -9
{{{9(4)+c = -3(-9)}}}
{{{36+c = 27}}}
{{{c = -9}}}
Check: Substitute a = 4, b = -9, and c = -9
First equation:
{{{y[1] = 4(x-3)^2+(-9)}}}
{{{y[1] = 4(x^2-6x+9)+(-9)}}}
{{{y[1] = 4x^2-24x+(36+(-9))}}}
{{{y[1] = 4x^2-24x+27}}}
Second equation:
{{{y[2] = (2x-3)(2x+(-9))}}}
{{{y[2] = (2x-3)(2x-9)}}}
{{{y[2] = 4x^2-18x-6x+27}}}
{{{y[2] = 4x^2-24x+27}}}
As you can see, the two equations are identical if:
a = 4, b = -9, and c = -9