Find discriminant, state roots

6

1. Calculate the discriminant of \(- 2 x ^ { 2 } + 7 x + 3\) and hence state the number of real roots of the equation \(- 2 x ^ { 2 } + 7 x + 3 = 0\).
2. The quadratic equation \(2 x ^ { 2 } + ( p + 1 ) x + 8 = 0\) has equal roots. Find the possible values of \(p\).

7

1. Calculate the discriminant of each of the following:
   1. \(x ^ { 2 } + 6 x + 9\),
   2. \(x ^ { 2 } - 10 x + 12\),
   3. \(x ^ { 2 } - 2 x + 5\).
   4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_391_446_628_397} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_449_625_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_389_442_630_1384} \captionsetup{labelformat=empty} \caption{Fig. 3}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_446_1119_644} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_396_447_1119_1137} \captionsetup{labelformat=empty} \caption{Fig. 5}
      \end{figure} State with reasons which of the diagrams corresponds to the curve
      (a) \(y = x ^ { 2 } + 6 x + 9\),
      (b) \(y = x ^ { 2 } - 10 x + 12\),
      (c) \(y = x ^ { 2 } - 2 x + 5\).

9 You are given that \(n , n + 1\) and \(n + 2\) are three consecutive integers.

1. Expand and simplify \(n ^ { 2 } + ( n + 1 ) ^ { 2 } + ( n + 2 ) ^ { 2 }\).
2. For what values of \(n\) will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer. Section B (36 marks)

2

1. Find the discriminant of the equation \(3 x ^ { 2 } - 2 x + 5 = 0\).
2. Use your answer to part (a) to find the number of real roots of the equation \(3 x ^ { 2 } - 2 x + 5 = 0\).

1 Four possible sketches of \(y = a x ^ { 2 } + b x + c\) are shown below.
Given \(b ^ { 2 } - 4 a c = 0\) and \(a , b\) and \(c\) are non-zero constants, which sketch is the only one that could possibly be correct? Tick ( \(\checkmark\) ) one box. A \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_383_303_995_550} \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1128_1009} B \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1562_1009} C \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_223_300_1868_548} \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_108_109_2001_1009} D \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_388_301_2305_549}
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Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]

Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]