1. Solve \[\sqrt{4x+13}-\sqrt{x+1}=\sqrt{12-x}.\]

How could we get rid of the square roots?

Squaring both sides of this will still leave a square root, but could we rearrange and square again?

If we do this, will our solutions all be valid?

  1. One root of the equation \[3x^3+14x^2+2x-4=0\] is rational. Obtain this root and complete the solution of the equation.

What does the question mean by ‘rational’?

We could write this root as \(\dfrac{a}{b}\), where the fraction is in lowest terms. What happens if we substitute this into the cubic and simplify?

Since \(a\) and \(b\) have no common factors, what does this tell us?

Can you find any ways to minimise the number of possibilities for \(a\) and \(b\) that need checking? Can we then check all the possibilities?