Quadratic equations

Solve x³ + 10x² + 8x = 0 Where appropriate give your answers in surd form. Simplify your answers where possible.

All lengths shown are in centimetres.

Use Pythagoras’ theorem to show that 4w² – 8w – 21 = 0

Work out the size of angle x.

Solve by factorising 2x³ – 9x² – 5x = 0

Write 2x² – 16x + 7 in the form k(x + m)² + n where k, m and n are integers.

Solve (x – 1)² – 5 = 0

The equation of a circle is (x – 2)² + (y + 3)² = 16 The equation of a line is y = 4 – x The circle and line intersect at two points, A and B.

Show that the x-coordinates of A and B satisfy the equation 2x² – 18x + 37 = 0

For A, the x-coordinate and y-coordinate are both positive.

Work out the coordinates of A. Give each coordinate to 2 decimal places.

(2x – 3) is a factor of 6x³ – 25x² + 28x – 6

Give all solutions as exact values.

A curve has equation y = x³ – 5x² At two points on the curve, the rate of change of y with respect to x is 4

Work out an equation, in terms of x, to represent this information. Give your answer in the form ax² + bx + c = 0 where a, b and c are integers.

Hence, work out the two possible values of x. Give your answers to 3 significant figures.

f(x) = 200x³ + 100x² – 18x – 9

Use the factor theorem to show that (2 x + 1) is a factor of f(x).

Hence solve f(x) = 0

Here is a triangle.

Use the cosine rule to work out the value of x.

A function f is given by

A sketch of y = f(x) is shown.

Work out all the values of x for which f(x) = – 12

3x² + 2bx + 8a can be written in the form 3(x + a)² + b + 2

Work out the two possible pairs of values of a and b.