State whether the following quadratic equations have two distinct real roots. Justify your answer.
(i) \(x^2 - 3x + 4 = 0\)
(ii) \(2x^2 + x - 1 = 0\)
(iii) \(2x^2 - 6x + \dfrac{9}{2} = 0\)
(iv) \(3x^2 - 4x + 1 = 0\)
(v) \((x+4)^2 - 8x = 0\)
(vi) \((x - \sqrt{2})^2 - 2(x+1) = 0\)
(vii) \(\sqrt{2}x^2 - \dfrac{3}{\sqrt{2}}x + \dfrac{1}{\sqrt{2}} = 0\)
(viii) \(x(1-x) - 2 = 0\)
(ix) \((x-1)(x+2)+2=0\)
(x) \((x+1)(x-2)+x=0\)
(i) No (D = -7)
(ii) Yes (D = 9)
(iii) No (D = 0)
(iv) Yes (D = 4)
(v) No (D = 0)
(vi) Yes (D = 4)
(vii) Yes (D = 2)
(viii) Yes (D = 9)
(ix) Yes (D = 1)
(x) Yes (D = 5)
Key idea: For a quadratic equation \(ax^2 + bx + c = 0\), we calculate the discriminant:
\(D = b^2 - 4ac\)
a = 1, b = -3, c = 4
D = (-3)^2 - 4(1)(4) = 9 - 16 = -7
Since D < 0, there are no real roots.
a = 2, b = 1, c = -1
D = (1)^2 - 4(2)(-1) = 1 + 8 = 9
D > 0, so there are two distinct real roots.
a = 2, b = -6, c = 9/2
D = (-6)^2 - 4(2)(9/2) = 36 - 36 = 0
D = 0, so there are real roots but they are equal, not distinct.
a = 3, b = -4, c = 1
D = (-4)^2 - 4(3)(1) = 16 - 12 = 4
D > 0, so there are two distinct real roots.
Expand: (x+4)^2 = x^2 + 8x + 16
Equation: x^2 + 8x + 16 - 8x = x^2 + 16 = 0
a = 1, b = 0, c = 16
D = 0^2 - 4(1)(16) = -64
D < 0, so there are no real roots.
Expand: (x - √2)^2 = x^2 - 2√2x + 2
Equation: x^2 - 2√2x + 2 - 2x - 2 = x^2 - (2√2 + 2)x
a = 1, b = -(2√2 + 2), c = 0
D = b^2 - 4ac = (-(2√2+2))^2 = (2√2+2)^2
This is positive, so there are two distinct real roots.
Multiply whole equation by √2 to remove fractions:
2x^2 - 3x + 1 = 0
a = 2, b = -3, c = 1
D = (-3)^2 - 4(2)(1) = 9 - 8 = 1
D > 0, so two distinct real roots.
Expand: x - x^2 - 2 = 0
Rearrange: -x^2 + x - 2 = 0 → x^2 - x + 2 = 0
a = 1, b = -1, c = 2
D = (-1)^2 - 4(1)(2) = 1 - 8 = -7
D < 0, so no real roots.
Expand: (x-1)(x+2) = x^2 + x - 2
Add +2: x^2 + x - 2 + 2 = x^2 + x = 0
a = 1, b = 1, c = 0
D = (1)^2 - 4(1)(0) = 1
D > 0, so two distinct real roots.
Expand: (x+1)(x-2) = x^2 - x - 2
Add +x: x^2 - x - 2 + x = x^2 - 2
a = 1, b = 0, c = -2
D = (0)^2 - 4(1)(-2) = 0 + 8 = 8
D > 0, so two distinct real roots.