# Quadratic equations

You can always reduce a quadratic equation to this form, which is called general form:

Examples:

1) 3x^{2} - 5x + 2 = 0 a = 3 b = -5 c = 2

2) x^{2} – 3 = 0 a = 1 b = 0 c = -3

3) 2x^{2} – 100x = 0 a = 2 b = -100 c = 0

4) (x + 2) · (x – 3) = 14; x^{2} + 2x – 3x – 6 = 14

x^{2} + 2x – 3x – 6 – 14 = 0

x^{2} – x – 20 = 0; a = 1, b = -1, c = -20

The solutions to this one are x = 5 and x = -4

To solve a quadratic equation, we have to use the formula:

*Example 1:*

*Example 2:*

*Example 3:*

*Example 4:*

NOTE: The discriminant, Δ = b^{2} – 4ac, is the value which detemines the number of solutions: if Δ > 0 there are two solutions; if Δ = 0 there is one solution and if Δ < 0 there is no solution.

**INCOMPLETE EQUATIONS**

*Example 5: b = c = 0*

*Example 6: b = 0*

*Example 7: c = 0*

*Example 8: c = 0*

**Exercise**: solve the following quadratic equations:

a) x^{2} - 5x + 4 = 0

b) x^{2} - 18x + 81 = 0

c) x^{2} + x + 15 = 0

d) x^{2} - 7x = 0

e) 147x^{2} = 0

f) 4x^{2} - 9 = 0

Solutions: a) 1 and 4; b) 9; c) Φ; d) 0 and 7; e) 0; f) ±3/2

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