Express \(4x^2 - 12x + 13\) in the form \((2x + a)^2 + b\), where \(a\) and \(b\) are constants.

The function \(f\) is defined by \(f(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\). Express \(x^2 - 4x + 8\) in the form \((x-a)^2 + b\).

Express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), and \(c\) are constants.

Express \(x^2 + 6x + 2\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants.

The function \(f\) is defined for \(x \in \mathbb{R}\) by \(f(x) = x^2 + ax + b\), where \(a\) and \(b\) are constants. The solutions of the equation \(f(x) = 0\) are \(x = 1\) and \(x = 9\). Find:

1. the values of \(a\) and \(b\),
2. the coordinates of the vertex of the curve \(y = f(x)\).

Express \(2x^2 - 12x + 7\) in the form \(a(x+b)^2 + c\), where \(a, b\) and \(c\) are constants.

Express \(2x^2 - 10x + 8\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), and \(c\) are constants, and use your answer to state the minimum value of \(2x^2 - 10x + 8\).

Express \(4x^2 - 12x\) in the form \((2x + a)^2 + b\).

A curve is described by the equation \(y = 2x^2 - 3x\). Express \(2x^2 - 3x\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), and \(c\) are constants, and determine the coordinates of the vertex of the curve.

The function \(f\) is defined as \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\). Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\).

Rewrite the expression \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\). Then, find the coordinates of the minimum point on the curve.

Express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\).

Express \(2x^2 - 4x + 1\) in the form \(a(x + b)^2 + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2x^2 - 4x + 1\).

The equation of a curve is \(y = 4x^2 + 20x + 6\).

1. Express the equation in the form \(y = a(x + b)^2 + c\), where \(a, b,\) and \(c\) are constants.
2. Hence solve the equation \(4x^2 + 20x + 6 = 45\).
3. Sketch the graph of \(y = 4x^2 + 20x + 6\) showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the \(x\)- and \(y\)-axes.

(a) Express \(x^2 - 8x + 11\) in the form \((x + p)^2 + q\) where \(p\) and \(q\) are constants.

(b) Hence find the exact solutions of the equation \(x^2 - 8x + 11 = 1\).

Express \(2x^2 - 8x + 14\) in the form \(2[(x-a)^2 + b]\).

Rewrite the expression \(5y^2 - 30y + 50\) in the form \(5(y + a)^2 + b\), where \(a\) and \(b\) are constants.

Express \(16x^2 - 24x + 10\) in the form \((4x + a)^2 + b\).

Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants.

The equation of a curve is given by \(y = 2x^2 + kx + k - 1\), where \(k\) is a constant. Given that \(k = 2\), express the equation of the curve in the form \(y = 2(x + a)^2 + b\), where \(a\) and \(b\) are constants. Also, state the coordinates of the vertex of the curve.