Discriminant for real roots condition

6. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9 .$$

1. Determine the number of real roots that exist for the equation \(\mathrm { f } ( x ) = 0\).
2. Solve the equation \(\mathrm { f } ( x ) = 8\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.

\(x^2 + 2x + 3 \equiv (x + a)^2 + b\).

1. Find the values of the constants \(a\) and \(b\). [2]
2. Sketch the graph of \(y = x^2 + 2x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes. [3]
3. Find the value of the discriminant of \(x^2 + 2x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). [2]

The equation \(x^2 + kx + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Find the set of possible values of \(k\), giving your answer in surd form. [4]

\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]

Given that \(k = 4\),

1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]

\(\text{f}(x) = 4x^2 + 12x + 9\).

1. Determine the number of real roots that exist for the equation \(\text{f}(x) = 0\). [2]
2. Solve the equation \(\text{f}(x) = 8\), giving your answers in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are rational. [4]

$$f(x) = x^3 - (k+4)x + 2k,$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve with equation \(y = f(x)\) passes through the point \((2, 0)\). [1]
2. Find the values of \(k\) for which the equation \(f(x) = 0\) has exactly two distinct roots. [5]

Given that \(k > 0\), that the \(x\)-axis is a tangent to the curve with equation \(y = f(x)\), and that the line \(y = p\) intersects the curve in three distinct points,

1. find the set of values that \(p\) can take. [5]