color(blue)"Shortcut method - by sight")
Given -> y=x^2-3x-28 .......................................(1)
y=(x-3/2)^2-3/4-28
y=(x-3/2)^2-121/4
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color(purple)("Fuller explanation")
color(blue)("Step 1")
Write as" " y=(x^2-3x)-28
color(brown)("Divide the brackets contents by "x". These means that the right")color(brown)("hand side is no longer equal to "y)
y!=(x-3)-28
color(brown)("square the brackets")
y!=(x-3)^2-28
color(brown)("Halve the -3 from "(x-3))
y!=(x-3/2)^2-28
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color(blue)("Step 2")
color(brown)("Changing the equation so that it does equal "y)
Let a constant of correction be k then
y=(x-3/2)^2-28 + k...................................(2)
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color(blue)("Step 3")
color(brown)("To find the value of k")
color(green)("As equation (1) and equation (2) both equal y we can equate them") color(green)("to each other through y")
Equation (1) = y = Equation (2)
x^2-3x-28" "=" "(x-3/2)^2-28+k
cancel(x^2)-cancel(3x)-cancel(28)" "=" "cancel(x^2)-cancel(3x)+9/4-cancel(28)+k
k=-9/4......................................................(3)
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color(blue)("Step 4 - last move!")
color(brown)("Bringing it all together to give the final equation")
Substitute equation (3) into equation (2)
y=(x-3/2)^2-28 -9/4.
But -28-9/4 = -121/4 giving
color(green)(y=(x-3/2)^2-121/4.
