Bab 2 Persamaan Kuadratik

Posted on October 16, 2016 by user

2.6 Persamaan Kuadratik, SPM Praktis (Kertas 2)

Soalan 3:

Jika α dan β ialah punca-punca persamaan kuadratik 3x² + 2x– 5 = 0, bentukkan persamaan kuadratik yang mempunyai punca-punca yang berikut.

\begin{array}{l} (a) \frac{2}{α} dan \frac{2}{β} \\ (b) (α + \frac{2}{β}) dan (β + \frac{2}{α}) \end{array}

Penyelesaian:

3x² + 2x – 5 = 0

a = 3, b = 2, c = –5

Punca adalah α dan β.

\begin{array}{l} α + β = - \frac{b}{a} = - \frac{2}{3} \\ α β = \frac{c}{a} = \frac{- 5}{3} \end{array}

(a)

\begin{array}{l} Punca-punca yang baru ialah \frac{2}{α} dan \frac{2}{β} . \\ Hasil tambah punca-punca baru \\ = \frac{2}{α} + \frac{2}{β} = \frac{2 β + 2 α}{α β} \\ = \frac{2 (α + β)}{α β} = \frac{2 (- \frac{2}{3})}{- \frac{5}{3}} = \frac{4}{5} \end{array}

\begin{array}{l} Hasil darab punca-punca \\ = (\frac{2}{α}) (\frac{2}{β}) = \frac{4}{α β} \\ = \frac{4}{- \frac{5}{3}} = - \frac{12}{5} \end{array}

Guna rumus, x² – (hasil tambah punca)x + hasil darab punca = 0

Persamaan kuadratik yang baru ialah

x^{2} - (\frac{4}{5}) x + (- \frac{12}{5}) = 0

5x² – 4x– 12 = 0

(b)

\begin{array}{l} Punca-punca yang baru ialah (α + \frac{2}{β}) dan (β + \frac{2}{α}) . \\ Hasil tambah punca-punca baru \\ = (α + \frac{2}{β}) + (β + \frac{2}{α}) \end{array}

\begin{array}{l} = α + β + (\frac{2}{α} + \frac{2}{β}) = α + β + \frac{2 α + 2 β}{α β} \\ = α + β + \frac{2 (α + β)}{α β} \\ = \frac{4}{5} + \frac{2 (\frac{4}{5})}{- \frac{12}{5}} \\ = \frac{4}{5} - \frac{2}{3} = \frac{2}{15} \end{array}

\begin{array}{l} Hasil darab punca-punca \\ = (α + \frac{2}{β}) (β + \frac{2}{α}) \\ = α β + 2 + 2 + \frac{4}{α β} \end{array}

\begin{array}{l} = - \frac{12}{5} + 4 + \frac{4}{- \frac{12}{5}} \\ = - \frac{12}{5} + 4 - \frac{5}{3} = - \frac{1}{15} \end{array}

Persamaan kuadratik yang baru ialah

x^{2} - (\frac{2}{15}) x + (- \frac{1}{15}) = 0

15x² – 2x– 1 = 0