Prove that the straight line x + y = 1 touches the parabola $y = x - x^{2} .$

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Solution

straight line
$x + y = 1$
parabola
$y = x - x^{2}$
putting value of y in parabola eqn
$1 - x = x - x^{2}$
$x^{2} - 2 x + 1 = 0$
$(x - 1)^{2} = 0$
$x = 1$
we get only one value of x
it means given straight line touches the parabola

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