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Discriminant & Solutions

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Discriminant & Solutions DISCRIMINANT & SOLUTIONS Quadratic equations can be written in the form ��# + �� + � = 0. (To use discriminants, if you have a quadratic (parabolic) equation in the vertex form, � = � � − ℎ # + �, you need to set � = 0 ; multiply through and convert to standard form as above.) ./± /1.234 The quadratic formula gives you the roots (where the function becomes zero); � = . #3 The part under the square root sign is called the discriminant, because it tells you where these roots appear on a graph. There are only three possibilities; the discriminant is positive, zero, or negative. 1: Positive Discriminant. This gives 2 unequal solutions that are real; they pass through the x- axis. They may be rational (if a, b, and c are all rational) or irrational. Examples: ������ ����ℎ: �# − � − 2; � = 1 , � = −1, � = −2. �# − 4�� = 9 > 0. So we have two real roots; {-1, 2}. �������� ����ℎ: − �# − � + 2; � = −1, � = −1, � = 2. �# − 4�� = 9 > 0. So we have two real roots; {-2, 1}; they are different from the above equation because a and c are different. 2: Discriminant = Zero. This means there will be one repeated solution that is a real number. Graphically, it touches the x-axis but doesn’t cross it; although it touches in only one point, there are still two roots at that x value. Examples: ������ ����ℎ: �# − 2� + 1; � = 1 , � = −2, � = 1. �# − 4�� = 0. So we have a repeated root; {1, 1}. �������� ����ℎ: − �# − 2� − 1; � = −1, � = −2, � = −1. �# − 4�� = 0. So we have a repeated root; {-1, -1}; it’s different from the above equation because a and c are different. 3: Negative Discriminant. There are NO real solutions (because the graph doesn’t cross the x- axis), but there are two imaginary roots that are complex conjugates. The real part of the imaginary number will be the x coordinate. Examples: ������ ����ℎ: �# − � + 1; � = 1 , � = −1, � = 1. �# − 4�� = −3 < 0 I J I J So we have two imaginary roots; { − �, + � } # # # # �������� ����ℎ: − �# − 2� − 1; � = −1, � = −2, � = −2. �# − 4�� = −4 < 0 So we have two imaginary roots; {−1 − �, −1 + � }. Vertex: To find the coordinate for the vertex point, you can either put the equation into the � = � � − ℎ # + � form and use the fact that the vertex is (ℎ, �) or leave it in the standard form and use the fact that the x coordinate of the vertex is given ./ by . Then just plug that value into the original � = ��# + �� + � to find the y coordinate. #3 If a is positive, the parabola opens upward, and the vertex (ℎ, �) is a minimum. If a is negative, the parabola opens downward, and the vertex (ℎ, �) is a maximum. Grade “A” Tutoring Page 1 GAT-KCC Discriminants-01a .
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