Complete square then solve equation

$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b ,$$ where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\).
2. Hence, or otherwise, show that the roots of $$x ^ { 2 } - 8 x - 29 = 0$$ are \(c \pm d \sqrt { } 5\), where \(c\) and \(d\) are integers to be found.

1. Solve the equation

$$x ^ { 2 } - 4 x - 8 = 0$$ giving your answers in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.

4. (i) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$ (ii) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$

1. The \(n\)th term of a sequence is defined by

$$u _ { n } = n ^ { 2 } - 6 n + 11 , \quad n \geq 1 .$$ Given that the \(k\) th term of the sequence is 38 , find the value of \(k\).

\(x^2 - 8x - 29 = (x + a)^2 + b\), where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\). [3]
2. Hence, or otherwise, show that the roots of $$x^2 - 8x - 29 = 0$$ are \(c \pm d\sqrt{5}\), where \(c\) and \(d\) are integers to be found. [3]

$$\text{f}(x) = 2x^2 - 4x + 1.$$

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\text{f}(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = \text{f}(x)\). [1]
3. Solve the equation \(\text{f}(x) = 3\), giving your answers in exact form. [3]

\(f(x) = 2x^2 - 4x + 1\).

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$f(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = f(x)\). [1]
3. Solve the equation \(f(x) = 3\), giving your answers in exact form. [3]