June 06, 2010, at 08:00 AM by LFS -
Changed line 13 from:

The formula is: x_{1,2} = {{ - b \,\pm\, \sqrt {b^2 \,-\, 4 \cdot a \cdot c} } \over {2 \cdot a}}

to:

The formula is: x_1,x_2 = {{ - b \,\pm\, \sqrt {b^2 \,-\, 4 \cdot a \cdot c} } \over {2 \cdot a}}

June 06, 2010, at 07:59 AM by LFS -
Changed line 13 from:

The formula is: \bbox[border:2px green dotted,2pt]{x_{1,2} = {{ - b \,\pm\, \sqrt {b^2 \,-\, 4 \cdot a \cdot c} } \over {2 \cdot a}}}

to:

The formula is: x_{1,2} = {{ - b \,\pm\, \sqrt {b^2 \,-\, 4 \cdot a \cdot c} } \over {2 \cdot a}}

March 15, 2008, at 10:40 PM by LFS -
Changed line 87 from:

[c] x_1=2

to:

[c] x_1=x_2=2

Changed line 99 from:
• The Quadratic formula and complex numbers
to:
• Quadratic formula and complex numbers
March 15, 2008, at 10:38 PM by LFS -
Changed line 13 from:

The formula is:   \bbox[border:2px green dotted,2pt]{x_{1,2} = {{ - b \,\pm\, \sqrt {b^2 - 4 \cdot a \cdot c} } \over {2 \cdot a}}}

to:

The formula is: \bbox[border:2px green dotted,2pt]{x_{1,2} = {{ - b \,\pm\, \sqrt {b^2 \,-\, 4 \cdot a \cdot c} } \over {2 \cdot a}}}

Changed line 21 from:

The discriminant is:    D = b^2 - 4 \cdot a \cdot c   (the expression inside the square root).

to:

The discriminant is:   D=b^2-4\cdot a\cdot c   (the expression inside the square root).

Changed line 26 from:

Solve the equation   x^2-2x-3=0    for x.

to:

Solve the equation: x^2-2x-3=0 for x.

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to:

Changed line 32 from:

x_1 = {{2 + 4} \over 2} = {6 \over 2} = 3  and  x_2 = {{2 - 4} \over 2} = {{ - 2} \over 2} = - 1

to:

x_1 = {{2 + 4} \over 2} = {6 \over 2} = 3 and x_2 = {{2 - 4} \over 2} = {{ - 2} \over 2} = - 1

Changed line 34 from:

Answer:    x_1=3    and    x_2=-1    are the two solutions to the equation   f(x)=x^2-2x-3=0     (see the left graph below!).

to:

Answer:   x_1=3 and x_2=-1 are the two solutions to the equation f(x)=x^2-2x-3=0 (see the left graph below!).

Changed lines 46-47 from:
• If the discriminant is positive, the formula gives two distinct numbers: x_1    и    x_2 .
In the above example, the discriminant was 16 (positive) and so the quadratic function f(x)=x^2-2x-3 has two distinct roots. This means the graph of the function has two x-intercepts x_1=3    and    x_2=-1 . See left graph below.
to:
• If the discriminant is positive, the formula gives two distinct numbers: x_1 and x_2 .

In the above example, the discriminant was 16 (positive) and so the quadratic function f(x)=x^2-2x-3 has two distinct roots. This means the graph of the function has two x-intercepts x_1=3 and x_2=-1 .  See left graph below.

Changed line 86 from:

[c] x_1=3    and    x_2=-1

to:

[c] x_1=3   and   x_2=-1

Changed line 88 from:

[c] x_1   and   x_2 do not exist!

to:

[c] x_1 and x_2 do not exist!

March 14, 2008, at 01:54 AM by LFS -
• The Quadratic formula and complex numbers
March 14, 2008, at 01:50 AM by LFS -
Changed line 99 from:
• root of a function  (x-intercepts)
to:
Changed line 101 from:
to:
March 14, 2008, at 01:27 AM by LFS -
Changed line 47 from:

In the above example, the discriminant was 16 and so the quadratic function f(x)=x^2-2x-3 has two distinct roots. This means the graph of the function has two x-intercepts x_1=3    and    x_2=-1 . See left graph below.

to:

In the above example, the discriminant was 16 (positive) and so the quadratic function f(x)=x^2-2x-3 has two distinct roots. This means the graph of the function has two x-intercepts x_1=3    and    x_2=-1 . See left graph below.

Changed line 64 from:

[c]

to:

[c]

March 13, 2008, at 11:21 PM by LFS -
Changed line 32 from:

x_1 = {{2 + 4} \over 2} = {6 \over 2} = 3  and  x_2 = {{2 - 4} \over 2} = {{ - 2} \over 2} = - 1

to:

x_1 = {{2 + 4} \over 2} = {6 \over 2} = 3  and  x_2 = {{2 - 4} \over 2} = {{ - 2} \over 2} = - 1

Changed line 34 from:

Answer:    x_1=3    and    x_2=-1    are the two solutions to the equation   f(x)=x^2-2x-3=0     (see the left graph below!).

to:

Answer:    x_1=3    and    x_2=-1    are the two solutions to the equation   f(x)=x^2-2x-3=0     (see the left graph below!).

Changed line 57 from:

[c] The discriminant D<0

to:

[c] The discriminant D<0

Changed line 61 from:

[c] D<0 - the parabola doesn't cross or touch the x-axis

to:

[c] D<0 - the parabola doesn't cross or touch the x-axis

Changed line 63 from:

[c]

to:

[c]

Changed line 72 from:

f(x)=2x^2+1

to:

f(x)=2x^2+1

Changed line 80 from:

[c] D= ( 0)^2 - 4 \cdot 2 \cdot 2=-8<0

to:

[c] D= ( 0)^2 - 4 \cdot 2 \cdot 2=-8<0

March 13, 2008, at 11:18 PM by LFS -
Changed line 34 from:

Answer:    x_1=3    and    x_2=-1    are the two solutions to the equation   f(x)=x^2-2x-3=0     (see the left graph below!).

to:

Answer:    x_1=3    and    x_2=-1    are the two solutions to the equation   f(x)=x^2-2x-3=0     (see the left graph below!).

Changed lines 63-64 from:

[c]

[c]

to:

[c]

[c]