IB Mathematics AA HL Flashcards- Solving equations

For these exams, you must be versatile in solving all types of equations. You find info on types such as: log, trig, and quadratic equations in other areas of these notes. I will just explain one non-calc. technique here:

[q]

HIDDEN QUADRATICS: So, in (2.7), we solved quadratics that look like this: \( ax^2 + bx + c = 0 \). However, \( x \) could be replaced with a whole other function, such as: \( 3e^{2x} – e^x + 12 = 0 \) or \( -2\cos^2x + 4\cos x – 9 = 0 \).

[a]

The process is:

– Replace the inner function with \( X \), write as a regular quadratic.
– Solve as normal to find what \( X \) equals.
– Set the inner function equal to those solutions & solve for the final solutions.

[q]
E.G.: Solve \( 12\sin x + 6 = \frac{6}{\sin x}, 0 < x < 2\pi \):

Write as \( 12 \sin x + 6 – \frac{6}{\sin x} = 0 \) → \( x = \sin x \).

\( 12 \sin^2 x + 6 \sin x – 6 = 0 \) → factor \( 6(2X^2 + X – 1) = 0 \).

Solve: \( X = \frac{1}{2}, X = -1 \). \( \sin x = \frac{1}{2} \), \( \sin x = -1 \).

\( x = \frac{\pi}{6} \) & \( \frac{3\pi}{2} \).

[a]

PAPER 2 SOLVES: Once or twice in every paper 2, there will be a question about any topic that will be way beyond the difficulty that our manual solving tools can handle. This means we should turn to our GDC.

[q]

We have two options when doing this:

1. Graph both sides of the equation and find the intersection (see 2.4).
2. Move everything to one side and find the zeros [usually easier/quicker].

[x] Exit text