How do I calculate algebric quadratic equations using trignotmetry with example.

Quadratic equations are solved algebraically; trigonometry is not required for the standard form \(ax^{2}+bx+c=0\).

Standard algebraic method (quadratic formula)

1. Write the equation in the form \(ax^{2}+bx+c=0\) where \(a\neq0\).
2. Identify the coefficients \(a,\;b,\;c\).
3. Substitute them into the quadratic formula \[ x=\frac{-b\pm\sqrt{\,b^{2}-4ac\,}}{2a} \] and simplify.

Example

Solve \(3x^{2}-7x+2=0\).

1. Here \(a=3,\;b=-7,\;c=2\).
2. Compute the discriminant: \[ \Delta=b^{2}-4ac=(-7)^{2}-4(3)(2)=49-24=25. \]
3. Apply the formula: \[ x=\frac{-(-7)\pm\sqrt{25}}{2(3)} =\frac{7\pm5}{6}. \]
4. Hence \[ x_{1}=\frac{7+5}{6}=\frac{12}{6}=2,\qquad x_{2}=\frac{7-5}{6}=\frac{2}{6}=\frac13. \]

Solutions: \(x=2\) and \(x=\frac13\)

Common exam mistake

Students often forget the sign of \(b\) when substituting into \(-b\). Remember that \(-b\) means “the opposite of the coefficient \(b\)”, not “the negative of the whole term”. In the example above, \(b=-7\) so \(-b=+7\).

When is trigonometry useful?

Trigonometric identities are employed for solving equations that contain both algebraic and trigonometric terms, e.g., \( \sin x = 2x^{2}+3x-1\). For a pure quadratic, the algebraic methods above are the most efficient and are the ones asked in ICSE/CBSE exams.