# How Many Roots? Examples

### Example 1

How many real number or complex number roots does the following equation have? *
y* = *x*^{2} + 4*x* + 1
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There's an easier way to find out than going through the trouble of finding all the roots yourself. Let's use the discriminate! *b*^{2} – 4*ac*. Er…we might need to know what *a*, *b*, and *c* are first. *a* = 1
*b* = 4
*c* = 1
16 – 4(1)(1) = 16 – 4 = 12 The discriminate is positive, so there are two real number roots for this equation. | |

### Example 2

How many real number or complex number roots does the following equation have? *
y* = *x*^{2} – *x* +3
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Let's start off with the set up. *a* = 1
*b* = -1
*c* = 3
And now we plug away. *b*^{2} – 4*ac* = (-1)^{2} – 4(1)(3) = 1 – 12 = -11
This time the discriminate is negative, so there are two complex number roots (and no real number roots). | |

### Example 3

How many real number or complex number roots does the following equation have? *y* = -2*x*^{2} – 4*x* + 1
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*a* = -2
*b* = -4
*c* = 1
*b*^{2} – 4*ac* = (-4)^{2} – 4(-2)(1) = 12 + 8 = 20
A positive outcome leads to real results—and two real roots. | |

### Example 4

How many real number or complex number roots does the following equation have? *y* = -9*x*^{2} + 6*x* – 1
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*a* = -9
*b* = 6
*c* = -1
*b*^{2} – 4*ac* = (6)^{2} – 4(-9)(-1) = 36 – 36 = 0
The discriminate…it's nothing. A nihilist's dream. In the end, there is only one (real root). | |

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