Write as:" "x^2+4x-21=0 x2+4x−21=0...................(1)
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color(blue)("Shortcut method")Shortcut method
By sight:
(x+2)^2-4-21=0(x+2)2−4−21=0
color(blue)((x+2)^2-25=0)(x+2)2−25=0
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color(blue)("The logic behind the shortcut method")The logic behind the shortcut method
color(green)("Consider only the left hand side")Consider only the left hand side
color(green)("Place the two left most terms in brackets")Place the two left most terms in brackets
" "(x^(color(magenta)(2)) +4x) -21 (x2+4x)−21
color(green)("Move the index of "color(magenta)(2)" to the outside of the brackets")Move the index of 2 to the outside of the brackets
" "(x+4color(red)(x))^(color(magenta)(2))-21 (x+4x)2−21
color(green)("Remove the "color(red)(x)" from the "4x" inside the brackets")Remove the x from the 4x inside the brackets
" "(x+color(blue)(4))^2-21 (x+4)2−21
color(green)("Apply "(1/2)xxcolor(blue)(4)=color(red)(2))Apply (12)×4=2
" "(x+color(red)(2))^2-21 (x+2)2−21
color(green)("Add the constant "color(red)(k))Add the constant k
" "(x+2)^2+color(red)(k)-21 (x+2)2+k−21........................(2)
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color(brown)("Determine the value of "k)Determine the value of k
Set equation (1) = equation (2)
" "x^2+4x-21=(x+2)^2+k-21 x2+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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