1.02d Quadratic functions: graphs and discriminant conditions

13. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
2. Find the range of possible values for \(k\).
   

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4. The equation \(( p - 2 ) x ^ { 2 } + 8 x + ( p + 4 ) = 0 , \quad\) where \(p\) is a constant has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } + 2 p - 24 > 0\)
2. Hence find the set of possible values of \(p\).

7. The equation \(2 x ^ { 2 } + 5 p x + p = 0\), where \(p\) is a constant, has no real roots. Find the set of possible values for \(p\).

10. The equation $$k x ^ { 2 } + 4 x + k = 2 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 4 < 0$$
2. Hence find the set of all possible values of \(k\).

9. The equation \(x ^ { 2 } + ( 6 k + 4 ) x + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$9 k ^ { 2 } + 12 k + 1 < 0$$
2. Find the range of possible values for \(k\), giving your boundaries as fully simplified surds.

8. The equation \(( k - 4 ) x ^ { 2 } - 4 x + k - 2 = 0\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$k ^ { 2 } - 6 k + 4 > 0$$
2. Find the exact range of possible values for \(k\).

11. The straight line \(l\) has equation \(y = m x - 2\), where \(m\) is a constant. The curve \(C\) has equation \(y = 2 x ^ { 2 } + x + 6\) The line \(l\) does not cross or touch the curve \(C\).

1. Show that \(m\) satisfies the inequality $$m ^ { 2 } - 2 m - 63 < 0$$
2. Hence find the range of possible values of \(m\).

11. The equation \(5 x ^ { 2 } + 6 = k \left( 13 x ^ { 2 } - 12 x \right)\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } + 13 k - 5 > 0$$
2. Find the set of possible values for \(k\).

11. The equation \(7 x ^ { 2 } + 2 k x + k ^ { 2 } = k + 7\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } - 7 k - 49 < 0$$
2. Find the range of possible values for \(k\).

8. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
2. Hence find the set of possible values of \(k\).

3. Given that the equation \(k x ^ { 2 } + 12 x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\).

10. $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$

1. Find the values of the constants \(a\) and \(b\).
2. In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
3. Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
4. Find the set of possible values of \(k\), giving your answer in surd form.

5. The equation \(2 x ^ { 2 } - 3 x - ( k + 1 ) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\).

8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for \(x\).

1. Show that \(k\) satisfies \(k ^ { 2 } + 4 k - 32 < 0\).
2. Hence find the set of possible values of \(k\).

7. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0 .$$
2. Hence find the set of possible values of \(k\).

10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
2. find the set of possible values of \(k\). Given that \(k = 1\),
3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.

8. The equation \(x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies $$k ^ { 2 } + 2 k - 3 > 0$$
2. Find the set of possible values of \(k\).

9. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
2. Hence find the set of possible values of \(k\).

8. The equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\), where \(p\) is a positive constant, has equal roots.

1. Find the value of \(p\).
2. For this value of \(p\), solve the equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\).

7. The equation \(x ^ { 2 } + k x + ( k + 3 ) = 0\), where \(k\) is a constant, has different real roots.

1. Show that \(k ^ { 2 } - 4 k - 12 > 0\).
2. Find the set of possible values of \(k\).

Given that the equation \(2 q x ^ { 2 } + q x - 1 = 0\), where \(q\) is a constant, has no real roots,

1. show that \(q ^ { 2 } + 8 q < 0\).
2. Hence find the set of possible values of \(q\).

6. The equation \(x ^ { 2 } + 3 p x + p = 0\), where \(p\) is a non-zero constant, has equal roots. Find the value of \(p\).

4. (a) Show that \(x ^ { 2 } + 6 x + 11\) can be written as $$( x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers to be found.
(b) In the space at the top of page 7 , sketch the curve with equation \(y = x ^ { 2 } + 6 x + 11\), showing clearly any intersections with the coordinate axes.
(c) Find the value of the discriminant of \(x ^ { 2 } + 6 x + 11\)

7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where \(k\) is a real constant.

1. Find the discriminant of \(\mathrm { f } ( x )\) in terms of \(k\).
2. Show that the discriminant of \(\mathrm { f } ( x )\) can be expressed in the form \(( k + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
3. Show that, for all values of \(k\), the equation \(\mathrm { f } ( x ) = 0\) has real roots.