Find range for two distinct roots

3 The equation of a curve is \(y = 2 x ^ { 2 } + m ( 2 x + 1 )\), where \(m\) is a constant, and the equation of a line is \(y = 6 x + 4\). Show that, for all values of \(m\), the line intersects the curve at two distinct points.

1 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 k x + k = 0\) has distinct real roots.

12

1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).

1. Find the set of values of the constant \(k\) such that the equation

$$x ^ { 2 } - 6 x + k = 0$$ has real and distinct roots.

1 Determine the set of values of \(k\) such that the equation \(x ^ { 2 } + 4 x + ( k + 3 ) = 0\) has two distinct real roots.

13 In this question you must show detailed reasoning. Determine the values of \(k\) for which part of the graph of \(y = x ^ { 2 } - k x + 2 k\) appears below the \(x\)-axis.

3 The equation \(k x ^ { 2 } + ( k - 6 ) x + 2 = 0\) has two distinct real roots. Find the set of possible values of the constant \(k\), giving your answer in set notation.

2 Let \(\mathrm { f } ( x ) = x ^ { 2 } + k x + 4\), where \(k\) is a constant.

1. Find an expression for the discriminant of f in terms of \(k\).
2. Hence find the range of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has two distinct real roots.

A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form. [5 marks]

The quadratic equation \(3x^2 + 4x + (2k - 1) = 0\) has real and distinct roots. Find the possible values of the constant \(k\) Fully justify your answer. [4 marks]

A curve, C, has equation \(y = x^2 - 6x + k\), where \(k\) is a constant. The equation \(x^2 - 6x + k = 0\) has two distinct positive roots.

1. Sketch C on the axes below. [2 marks]
2. Find the range of possible values for \(k\). Fully justify your answer. [4 marks]

Find all the values of \(k\) for which the equation \(x^2 + 2kx + 9k = -4x\) has two distinct real roots. [7]

It is given that $$f(x) = x^2 - kx + (k+3),$$ where \(k\) is a constant. If the equation \(f(x) = 0\) has real roots find the range of the possible values of \(k\). [4]

In this question you must show detailed reasoning. A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form using set notation. [6]

The equation \(kx^2 + 4x + (5 - k) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\). Find the set of possible values of \(k\). Write your answer using set notation. [5]