ESAT · Maths

ESAT 'how many solutions' questions

Count how many real solutions an equation has, often without solving it fully.

How to spot it

How many real solutions does ... have?
How many values of x in the range ... satisfy ...?

Try one

Original example

How many real solutions does the equation $x^4 - 3x^2 - 4$ $= 0$ have?

The method

Spot a substitution or factorisation that splits the equation.
Check each factor: can it actually be zero for a real x?
For trigonometry, count over the full range of the inner angle.

The trap

Watch forCounting every factor's roots, including ones with no real solution.

Now a harder one

Harder

How many values of $x$ with $0^\circ$ $\leq x$ $< 360^\circ$ satisfy $\sin 2x$ $= \frac{1}{2}$ ?

More ESAT question types

Which statements are correct Scaling by a factor Reacting masses