Question:

1. General method – For each quadratic we either factorise (if possible) or use the quadratic formula \[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] After obtaining the two zeroes $\alpha$ and $\beta$, compute \(\alpha+\beta\) and \(\alpha\beta\) and compare with \(-b/a\) and \(c/a\) respectively.\
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(i) $x^2-2x-8$\
- $a=1,\;b=-2,\;c=-8$.\
- Factorising: $x^2-2x-8=(x-4)(x+2)$.\
- Zeroes: $\alpha=4$, $\beta=-2$.\
- Sum $\alpha+\beta = 4+(-2)=2 = -\frac{b}{a}= -\frac{-2}{1}=2$.\
- Product $\alpha\beta = 4\times(-2) = -8 = \frac{c}{a}=\frac{-8}{1}=-8$.\
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(ii) $4s^2-4s+1$\
- $a=4,\;b=-4,\;c=1$.\
- Discriminant $\Delta = (-4)^2-4\cdot4\cdot1 = 0$.\
- Double root: $s = \frac{-b}{2a}=\frac{4}{8}=\frac12$.\
- Zeroes: $\alpha=\beta=\frac12$.\
- Sum $\alpha+\beta = 1 = -\frac{b}{a}= -\frac{-4}{4}=1$.\
- Product $\alpha\beta = \frac14 = \frac{c}{a}=\frac{1}{4}$.\
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(iii) $6x^2-7x-3$ (re‑ordered from $6x^2-3-7x$)\
- $a=6,\;b=-7,\;c=-3$.\
- $\Delta = (-7)^2-4\cdot6\cdot(-3)=49+72=121$.\
- Roots: $x = \frac{7\pm\sqrt{121}}{12}=\frac{7\pm11}{12}$.\
\[\alpha = \frac{18}{12}=\frac32,\qquad \beta = \frac{-4}{12}= -\frac13.\]\
- Sum $\alpha+\beta = \frac32-\frac13 = \frac{7}{6}= -\frac{b}{a}= -\frac{-7}{6}=\frac{7}{6}$.\
- Product $\alpha\beta = \frac32\times\left(-\frac13\right)= -\frac12 = \frac{c}{a}=\frac{-3}{6}= -\frac12$.\
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(iv) $4u^2+8u$\
- $a=4,\;b=8,\;c=0$.\
- Factor: $4u(u+2)=0$.\
- Zeroes: $\alpha=0$, $\beta=-2$.\
- Sum $\alpha+\beta = -2 = -\frac{b}{a}= -\frac{8}{4}= -2$.\
- Product $\alpha\beta = 0 = \frac{c}{a}=\frac{0}{4}=0$.\
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(v) $t^2-15$\
- $a=1,\;b=0,\;c=-15$.\
- Roots: $t = \pm\sqrt{15}$.\
- Zeroes: $\alpha=\sqrt{15}$, $\beta=-\sqrt{15}$.\
- Sum $\alpha+\beta = 0 = -\frac{b}{a}=0$.\
- Product $\alpha\beta = -15 = \frac{c}{a}= -15$.\
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(vi) $3x^2 - x - 4$\
- $a=3,\;b=-1,\;c=-4$.\
- $\Delta = (-1)^2-4\cdot3\cdot(-4)=1+48=49$.\
- Roots: $x = \frac{1\pm7}{6}$.\
\[\alpha = \frac{8}{6}=\frac43,\qquad \beta = \frac{-6}{6}= -1.\]\
- Sum $\alpha+\beta = \frac43-1 = \frac13 = -\frac{b}{a}= -\frac{-1}{3}=\frac13$.\
- Product $\alpha\beta = \frac43\times(-1)= -\frac43 = \frac{c}{a}=\frac{-4}{3}= -\frac43$.