Find k for equal roots

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 \(k\).

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 \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\).

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

1. Find the discriminant of \(k x ^ { 2 } - 4 x + k\) in terms of \(k\).
2. The quadratic equation \(k x ^ { 2 } - 4 x + k = 0\) has equal roots. Find the possible values of \(k\)

10 The quadratic equation \(k x ^ { 2 } - 30 x + 25 k = 0\) has equal roots. Find the possible values of \(k\).

1

1. Express \(2 x ^ { 2 } - 12 x + 23\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Use your result to show that the equation \(2 x ^ { 2 } - 12 x + 23 = 0\) has no real roots.
3. Given that the equation \(2 x ^ { 2 } - 12 x + k = 0\) has repeated roots, find the value of the constant \(k\).

4 The quadratic equation \(x ^ { 2 } + ( m + 4 ) x + ( 4 m + 1 ) = 0\), where \(m\) is a constant, has equal roots.

1. Show that \(m ^ { 2 } - 8 m + 12 = 0\).
2. Hence find the possible values of \(m\).

4. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c ,$$ where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).

1. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\).

You are given that \(\mathrm { f } ( x ) = 4 x ^ { 2 } + 12 x + c\), where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).

2

1. 1. Find the value of the discriminant of \(x ^ { 2 } + 3 x + 5\).
   2. Use your value from part (i) to determine the number of real roots of the equation \(x ^ { 2 } + 3 x + 5 = 0\).
2. Find the non-zero value of \(k\) for which the equation \(k x ^ { 2 } + 3 x + 5 = 0\) has only one distinct real root.

Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]

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

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

Find the values of \(k\) for which the equation \((2k - 3)x^2 - kx + (k - 1) = 0\) has equal roots. [4 marks]

The quadratic equation $$4x^2 + bx + 9 = 0$$ has one repeated real root. Find \(b\) Circle your answer. [1 mark] \(b = 0\) \quad \(b = \pm 12\) \quad \(b = \pm 13\) \quad \(b = \pm 36\)