Solve quadratic inequality

8

1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.

7 Solve the inequalities

1. \(3 - 8 x > 4\),
2. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\). \(8 \quad A\) is the point \(( - 2,6 )\) and \(B\) is the point \(( 3 , - 8 )\). The line \(l\) is perpendicular to the line \(x - 3 y + 15 = 0\) and passes through the mid-point of \(A B\). Find the equation of \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).

4 Solve the inequality \(x ( x - 6 ) > 0\).

4 Solve the following inequalities.

1. \(2 ( 1 - x ) > 6 x + 5\)
2. \(( 2 x - 1 ) ( x + 4 ) < 0\)

6 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).

2 Consider the two statements, A and B, below.
A: \(x ^ { 2 } - 6 x + 8 > 0\) B: \(x > 4\) Choose the most appropriate option below.
Circle your answer.
[0pt] [1 mark] $$A \Rightarrow B \quad A \Leftarrow B \quad A \Leftrightarrow B$$ There is no connection between \(A\) and B

1. In this question you should show all stages of your working.

Solutions relying on calculator technology are not acceptable.
Using algebra, solve the inequality $$x ^ { 2 } - x > 20$$ writing your answer in set notation.

6 In this question you must show detailed reasoning.

1. Solve the inequality \(x ^ { 2 } + x - 6 > 0\), giving your answer in set notation.
2. Solve the equation \(x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0\).
3. Find the exact solution of the equation \(\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }\).

1 Write the solution of the inequality \(( x - 2 ) ( x + 3 ) > 0\) using set notation.

14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers \(a\) and \(b\) \includegraphics[max width=\textwidth, alt={}]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-21_2491_1755_173_123} number Additional page, if required. Write the question numbers in the left-hand margin. \(\_\_\_\_\) number \section*{Additional page, if required. Write the question numbers in the left-hand margin.
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\hline & \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \hline & \begin{tabular}{l}

7 Solve each of the following inequalities:

1. \(3 ( x - 1 ) > 3 - 5 ( x + 6 )\);
2. \(\quad x ^ { 2 } - x - 6 < 0\).

7 Solve each of the following inequalities:

1. \(\quad 2 ( 4 - 3 x ) > 5 - 4 ( x + 2 )\);
2. \(\quad 2 x ^ { 2 } + 5 x \geqslant 12\).

8 Solve the following inequalities:

1. \(\quad 3 ( 1 - 2 x ) - 5 ( 3 x + 2 ) > 0\)
2. \(\quad 6 x ^ { 2 } \leqslant x + 12\) [0pt] [4 marks]

1. (a) Solve the inequality

$$3 x - 8 > x + 13$$ (b) Solve the inequality $$x ^ { 2 } - 5 x - 14 > 0$$

3. (a) Solve the inequality \(3 x - 8 > x + 13\).
(b) Solve the inequality \(x ^ { 2 } - 5 x - 14 > 0\).

2. Find the set of values of \(x\) for which $$( x - 1 ) ( x - 2 ) < 20$$

1. Solve the inequality

$$x ( 2 x + 1 ) \leq 6 .$$

1 In this question you must show detailed reasoning. Solve the inequality \(10 x ^ { 2 } + x - 2 > 0\).

7

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\).
2. Hence, or otherwise, solve \(2 x ^ { 2 } - x - 3 < 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of k .

17. (a) Solve the inequality $$3 x - 8 > x + 13$$ (b) Solve the inequality $$x ^ { 2 } - 5 x - 14 > 0$$

6 Solve the inequality \(x ^ { 2 } + 2 x < 3\).

6 Determine the set of values of \(x\) which satisfy the inequality $$3 x ^ { 2 } + 3 x > x + 6$$ Give your answer in exact form using set notation.
[0pt] [4 marks]

1 State the set of values of \(x\) which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\} \\ & \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\} \\ & \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\} \\ & \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$

1 The graph of \(y = a x ^ { 2 } + b x + c\) has roots \(x = 2\) and \(x = 5\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_905_963_717_625} State the set of values of \(x\) which satisfy $$a x ^ { 2 } + b x + c > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \{ x : x < 2 \} \cup \{ x : x > 5 \} \\ & \{ x : 0 < x < 2 \} \cap \{ x : x > 5 \} \\ & \{ x : 2 < x < 5 \} \\ & \{ x : 2 > x > 5 \} \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_118_115_1950_1087}
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