Write as:#" "x^2+4x-21=0#...................(1)
'~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Shortcut method")#
By sight:
#(x+2)^2-4-21=0#
#color(blue)((x+2)^2-25=0)#
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#color(blue)("The logic behind the shortcut method")#
#color(green)("Consider only the left hand side")#
#color(green)("Place the two left most terms in brackets")#
#" "(x^(color(magenta)(2)) +4x) -21#
#color(green)("Move the index of "color(magenta)(2)" to the outside of the brackets")#
#" "(x+4color(red)(x))^(color(magenta)(2))-21#
#color(green)("Remove the "color(red)(x)" from the "4x" inside the brackets")#
#" "(x+color(blue)(4))^2-21#
#color(green)("Apply "(1/2)xxcolor(blue)(4)=color(red)(2))#
#" "(x+color(red)(2))^2-21#
#color(green)("Add the constant "color(red)(k))#
#" "(x+2)^2+color(red)(k)-21#........................(2)
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#color(brown)("Determine the value of "k)#
Set equation (1) = equation (2)
#" "x^2+4x-21=(x+2)^2+k-21#
#" "cancel(x^2)+cancel(4x)-cancel(21)=cancel(x^2)+cancel(4x)+4+k-cancel(21)#
#" "k=-4#
So for #(x+b)^2+k+c" "-> k=-b^2#
#" "(x+2)^2-25=0" "........................(2_a)#
So #color(blue)("Vertex"->(x,y) =((-1)xx2,-25) = (-2,-25))#
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