Question

1) Calcule as raízes, caso existam, das seguintes funções quadráticas abaixo.

Answer 1

┃ -b +/- √∆
┃x = ──── ┃∆ = b^2 - 4ac
┃ 2a


b)

• y = x^2 + 3x + 2
{a = 1, b = 3, c = 2}

• ∆ = 3^2 - 4 . 1 . 2
• ∆ = 9 - 4 . 2
• ∆ = 9 - 8
• ∆ = 1

-3 +/- √1 -3 +/- 1
• ──── > ────
2 . 1 2

▊x1 = [-3 + 1]/2 > x1 = -2/2 > x1 = -1
▊x2= [-3 - 1]/2 > x2 = -4/2 > x2 = -2





c)

• y = x^2 + x - 2
{a = 1, b = 1, c = -2}

• ∆ = 1^2 - 4 . 1 . (-2)
• ∆ = 1 - 4 . (-2)
• ∆ = 1 + 8
• ∆ = 9

-1 +/- √9 -1 +/- 3
• ──── > ───
2 . 1 2

▊x1 = [-1 + 3]/2 > x1 = 2/2 > x1 = 1
▊x2 = [-1 - 3]/2 > x2 = -4/2 > x2 = -2





d)

• y = x^2 - 6x +9
{a = 1, b = -6, c = 9}

• ∆ = (-6)^2 - 4 . 1 . 9
• ∆ = 36 - 4 . 9
• ∆ = 36 - 36
• ∆ = 0

-(-6) +/- √0 6 +/- 0
• ───── > ───
2 . 1 2

▊x = 6/3 > x = 2





e)

• y = x^2 - 4x + 3
{a = 1, b = -4, c = 3}

• ∆ = (-4)^2 - 4 . 1 . 3
• ∆ = 16 - 4 . 3
• ∆ = 16 - 12
• ∆ = 4

-(-4) +/- √4 4 +/- 2
• ────── > ─── > 2 +/- 1
2 . 1 2

▊x1 = 2 + 1 > x1 = 3
▊x2 = 2 - 1 > 1





f)

• y = x^2 + 4x +3
{a = 1, b = 4, c = 3}

• ∆ = 4^2 - 4 . 1 . 3
• ∆ = 16 - 4 . 3
• ∆ = 16 - 12
• ∆ = 4

-4 +/- √4 -4 +/- 2
• ───── > ──── > -2 +/- 1
2 . 1 2

▊x1 = -2 + 1 > x1 = -1
▊x2 = -2 - 1 > x2 = -3





g)

• y = x^2 - x - 2
{a = 1, b = -1, c = -2}

• ∆ = (-1)^2 - 4 . 1 . (-2)
• ∆ = 1 - 4 . (-2)
• ∆ = 1 + 8
• ∆ = 9

-(-1) +/- √9 1 +/- 3
• ───── > ───
2 . 1 2

▊x1 = [1 + 3]/2 > x1 = 4/2 > x1 = 2
▊x2 = [1 - 3]/2 > x2 = -2/2 > x2 = -1





h)

• y = x^2 - 2x - 3
{a = 1, b = -2, c = -3}

• ∆ = (-2)^2 - 4 . 1 . (-3)
• ∆ = 4 - 4 . (-3)
• ∆ = 4 + 12
• ∆ = 16

-(-2) +/- √16 2 +/- 4
• ────── > ─── > 1 +/- 2
2 . 1 2

▊x1 = 1 + 2 > x1 = 3
▊x2 = 1 - 2 > x2 = -1