M151Q Precalculus · Unit 4: Trigonometric functions · on Final Exam
Quadratic-in-form Quadratic-in-form equations, trigonometric and algebraic
Study notes for Quadratic-in-form equations, trigonometric and algebraic in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- An equation is quadratic in form when a substitution $u = (\text{something})$ turns it into $au^2 + bu + c = 0$: $x^4 - 5x^2 + 4 = 0$ with $u = x^2$; $x^{2/3} - x^{1/3} - 6 = 0$ with $u = x^{1/3}$; $2\sin^2\theta - \sin\theta - 1 = 0$ with $u = \sin\theta$.
- Solve for $u$ by factoring or the quadratic formula, then undo the substitution for each $u$ value and keep only valid results ($|\sin\theta| \le 1$, radicands non-negative, domain of the original equation).
- Trig versions may need an identity first to get a single function: $2\cos^2\theta + \sin\theta = 2$ becomes $2(1 - \sin^2\theta) + \sin\theta = 2$.
- Report trig solutions for the required interval, using reference angles, and check for solutions where you divided by something.
Key formulas
Substitution
$$x^4 - 5x^2 + 4 = 0 \xrightarrow{u = x^2} u^2 - 5u + 4 = 0 \Rightarrow u = 1, 4 \Rightarrow x = \pm1, \pm2$$
Trig form
$$2\sin^2\theta - \sin\theta - 1 = 0 \Rightarrow (2\sin\theta + 1)(\sin\theta - 1) = 0$$
Worked example
Solve $2\sin^2\theta - \sin\theta - 1 = 0$ for $0^\circ \le \theta < 360^\circ$, and $x - 5\sqrt x + 6 = 0$.
Show the solution
$(2\sin\theta + 1)(\sin\theta - 1) = 0$: $\sin\theta = -\tfrac12 \Rightarrow 210^\circ, 330^\circ$; $\sin\theta = 1 \Rightarrow 90^\circ$. With $u = \sqrt x$: $u^2 - 5u + 6 = 0 \Rightarrow u = 2, 3 \Rightarrow x = 4, 9$ (both check).
Common mistakes
- Stopping at the u-values.
- Accepting sin θ = 2 as a solution.
- Losing the negative root when u = x².
Exam tipWrite "let u = …" explicitly on the exam. It shows the method and keeps you from forgetting to substitute back.
Keep going in Mathub
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