Question

Write a quadratic equation that goes through the points (2,3), (3,2), and (4,3).

y = ax2 + bx + c

Answer 1

`\sf\\\textsf{We have,}\\y=ax^2+bx+c.......(1)\\`

`\sf\\\textsf{Putting (2,3) in equation(1),}\\y=a(2)^2+b(2)+c\\or,\ 3=4a+2b+c.......(2)`

`\sf\\\textsf{Putting (3,2) in equation(1),}\\y=a(3)^2+b(3)+c\\or,\ 2=9a+3b+c.....(3)`

`\sf\\\textsf{Putting (4,3) in equation(1),}\\y=a(4)^2+b(4)+c\\or,\ 3=16a+4b+c....(4)`

`\sf\\\textsf{Subtracting equation(2) from equation(4),}\\0=12a+2b\\or,\ -2b=12a\\or,\ b=-6a......(5)`

`\sf\\\textsf{Substituting b = -6a in equation(3),}\\2=9a+3(-6a)+c\\or,\ 2=-9a+c\\or,\ c=9a+2.......(6)`

`\sf\\\textsf{Substituting values of b and c in equation(2),}\\3=4a+2(-6a)+9a+2\\or,\ 3=4a-12a+9a+2\\or,\ 3=a+2\\or,\ a=1`

`\sf\\\textsf{Equation(5) becomes: }\\b=-6(1)=-6`

`\sf\\\textsf{Equation(6) becomes: }\\c=9(1)+2\\or,\ c=11`

`\sf\\\therefore\ \textsf{Equation(1) becomes: }\\y=1x^2+(-6)x+11\\or,\ y=x^2-6x+11\textsf{ is the required quadratic equation.}`