## GlossaryT.QuadraticFormula History

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

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

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}}}

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

[c] x_1=2

[c] x_1=x_2=2

- The Quadratic formula and complex numbers

- Quadratic formula and complex numbers

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}}}

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}}}

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

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

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

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

Solution: Here а=1 \quad b=-2 \quad c=-3 (other examples?)

Solution: Here а=1 \quad b=-2 \quad c=-3 (other examples?)

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

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

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!).

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!).

- 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.

- 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.

[c] x_1=3 and x_2=-1

[c] x_1=3 and x_2=-1

[c] x_1 and x_2 do not exist!

[c] x_1 and x_2 do not exist!

- The Quadratic formula and complex numbers

- root of a function (x-intercepts)

- root of a function, x-intercept

- linear function, polynom? (polynomial function)

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.

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.

[c]

[c]

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

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

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!).

[c] The discriminant D<0

[c] The discriminant D<0

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

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

[c]

[c]

f(x)=2x^2+1

f(x)=2x^2+1

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

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

[c]

[c]

[c]

[c]