Completing the square

Express 3x² - 12x + 17 in the form a(x - b)² + c where a, b and c are constants.

State the coordinates of the minimum point of the curve y = 3x² - 12x + 17.

State the equation of the normal to the curve y = 3x² - 12x + 17 at its minimum point.

Express x² - 8x + 11 in the form (x - a)² + b where a and b are constants.

Hence write down the minimum value of x² - 8x + 11.

A circle has centre C which lies on the x-axis, as shown in the diagram. The line y = x meets the circle at A and B. The midpoint of AB is M.

The equation of the circle is x² - 6x + y² + a = 0, where a is a constant.

In this question you must show detailed reasoning.

Show that the area of triangle ABC is

Find the value of a when the area of triangle ABC is zero.

Give a geometrical interpretation of the case in part (b)(i).

Give a geometrical interpretation of the case where a = 5.

Write 2x² + 6x + 7 in the form p(x + q)² + r, where p, q and r are constants.

State the coordinates of the minimum point on the graph of y = 2x² + 6x + 7.

Hence deduce

• the minimum value of 2 tan²θ + 6 tan θ + 7,

• the smallest positive value of θ, in degrees, for which the minimum value occurs.

The functions f and g are defined for all real values of x by

f(x) = 2x² + 6x and g(x) = 3x + 2.

Find the range of f.

Give a reason why f has no inverse.

Given that fg(-2) = g^-1(a), where a is a constant, determine the value of a.

Determine the set of values of x for which f (x) > g (x). Give your answer in set notation.

A circle with centre C has equation x² + y² - 6x + 4y + 4 = 0.

Find

the coordinates of C,

the radius of the circle.

Determine the set of values of k for which the line y = kx - 3 does not intersect or touch the

circle.

Express 2x² - 12x + 23 in the form a(x + b)² + c.

Use your result to show that the equation 2x² - 12x + 23 = 0 has no real roots.

A circle with centre C has equation x² +y² +8x-2y-7 = 0.

Find

the coordinates of C,

the radius of the circle.