Soalan 2:
Tulis setiap persamaan kuadratik yang berikut dalam bentuk am. Seterusnya selesaikan persamaan kuadratik tersebut.
Penyelesaian:
(a)
$$ \begin{aligned} &\begin{aligned} & m(m+2)=3 \\ & m^2+2 m=3 \\ & m^2+2 m-3=0 \\ &(m+3)(m-1)=0 \\ & m+3=0 \\ & m=-3 \end{aligned}\\ &\text { atau, } \quad m-1=0\\ &m=1 \end{aligned} $$
(b)
$$ \begin{aligned} &\begin{aligned} 3 p(11-2 p) & =15 \\ 33 p-6 p^2-15 & =0 \\ -6 p^2+33 p-15 & =0 \end{aligned}\\ &(\div 3)\\ &\begin{gathered} -2 p^2+11 p-5=0 \\ 2 p^2-11 p+5=0 \\ (2 p-1)(p-5)=0 \\ 2 p-1=0 \\ 2 p=1 \\ p=\frac{1}{2} \end{gathered}\\ &\text { atau, } p-5=0\\ &p=5 \end{aligned} $$
(c)
$$ \begin{aligned} \frac{1}{2} y^2 & =12-y \\ y^2 & =24-2 y \\ y^2+2 y-24 & =0 \\ (y-4)(y+6) & =0 \\ y=4 \quad \text { atau } & y=-6 \end{aligned} $$
(d)
$$ \begin{aligned} & a+\frac{5}{a}=6 \\ & a^2+5= 6 a \\ & a^2-6 a+5= 0 \\ &(a-5)(a-1)=0 \\ & a=5 \quad \text { atau } \quad a=1 \end{aligned} $$
(e)
$$ \begin{aligned} & \frac{8}{k}=2+k \\ & 8=2 k+k^2 \\ &-k^2-2 k+8=0 \\ & k^2+2 k-8=0 \\ &(k+4)(k-2)=0 \\ & k=-4 \quad \text { atau } \quad k=2 \end{aligned} $$
(f)
$$ \begin{aligned} & 2 h+\frac{6}{h}=7 \\ & 2 h^2+6=7 h \\ & 2 h^2-7 h+6=0 \\ &(2 h-3)(h-2)=0 \\ & h=\frac{3}{2} \quad \text { atau } \quad h=2 \end{aligned} $$
(g)
$$ \begin{aligned} (h-2)(h-1) & =12 \\ h^2-h-2 h+2-12 & =0 \\ h^2-3 h-10 & =0 \\ (h-5)(h+2) & =0 \\ h=5 \quad \text { atau } & h=-2 \end{aligned} $$
(h)
$$ \begin{aligned} &(2 x-1)^2=3 x-2 \\ &(2 x-1)(2 x-1)=3 x-2 \\ & 4 x^2-2 x-2 x+1=3 x-2 \\ & 4 x^2-2 x-2 x+1-3 x+2=0 \\ & 4 x^2-7 x+3=0 \\ &(4 x-3)(x-1)=0 \\ & x=\frac{3}{4} \text { atau } x=1 \end{aligned} $$
(i)
$$ \begin{aligned} (r+1)(r+9) & =16 r \\ r^2+9 r+1 r+9 & =16 r \\ r^2+9 r+r+9-16 r & =0 \\ r^2-6 r+9 & =0 \\ (r-3)(r-3) & =0 \\ r & =3 \end{aligned} $$
Tulis setiap persamaan kuadratik yang berikut dalam bentuk am. Seterusnya selesaikan persamaan kuadratik tersebut.
Penyelesaian:
(a)
$$ \begin{aligned} &\begin{aligned} & m(m+2)=3 \\ & m^2+2 m=3 \\ & m^2+2 m-3=0 \\ &(m+3)(m-1)=0 \\ & m+3=0 \\ & m=-3 \end{aligned}\\ &\text { atau, } \quad m-1=0\\ &m=1 \end{aligned} $$
(b)
$$ \begin{aligned} &\begin{aligned} 3 p(11-2 p) & =15 \\ 33 p-6 p^2-15 & =0 \\ -6 p^2+33 p-15 & =0 \end{aligned}\\ &(\div 3)\\ &\begin{gathered} -2 p^2+11 p-5=0 \\ 2 p^2-11 p+5=0 \\ (2 p-1)(p-5)=0 \\ 2 p-1=0 \\ 2 p=1 \\ p=\frac{1}{2} \end{gathered}\\ &\text { atau, } p-5=0\\ &p=5 \end{aligned} $$
(c)
$$ \begin{aligned} \frac{1}{2} y^2 & =12-y \\ y^2 & =24-2 y \\ y^2+2 y-24 & =0 \\ (y-4)(y+6) & =0 \\ y=4 \quad \text { atau } & y=-6 \end{aligned} $$
(d)
$$ \begin{aligned} & a+\frac{5}{a}=6 \\ & a^2+5= 6 a \\ & a^2-6 a+5= 0 \\ &(a-5)(a-1)=0 \\ & a=5 \quad \text { atau } \quad a=1 \end{aligned} $$
(e)
$$ \begin{aligned} & \frac{8}{k}=2+k \\ & 8=2 k+k^2 \\ &-k^2-2 k+8=0 \\ & k^2+2 k-8=0 \\ &(k+4)(k-2)=0 \\ & k=-4 \quad \text { atau } \quad k=2 \end{aligned} $$
(f)
$$ \begin{aligned} & 2 h+\frac{6}{h}=7 \\ & 2 h^2+6=7 h \\ & 2 h^2-7 h+6=0 \\ &(2 h-3)(h-2)=0 \\ & h=\frac{3}{2} \quad \text { atau } \quad h=2 \end{aligned} $$
(g)
$$ \begin{aligned} (h-2)(h-1) & =12 \\ h^2-h-2 h+2-12 & =0 \\ h^2-3 h-10 & =0 \\ (h-5)(h+2) & =0 \\ h=5 \quad \text { atau } & h=-2 \end{aligned} $$
(h)
$$ \begin{aligned} &(2 x-1)^2=3 x-2 \\ &(2 x-1)(2 x-1)=3 x-2 \\ & 4 x^2-2 x-2 x+1=3 x-2 \\ & 4 x^2-2 x-2 x+1-3 x+2=0 \\ & 4 x^2-7 x+3=0 \\ &(4 x-3)(x-1)=0 \\ & x=\frac{3}{4} \text { atau } x=1 \end{aligned} $$
(i)
$$ \begin{aligned} (r+1)(r+9) & =16 r \\ r^2+9 r+1 r+9 & =16 r \\ r^2+9 r+r+9-16 r & =0 \\ r^2-6 r+9 & =0 \\ (r-3)(r-3) & =0 \\ r & =3 \end{aligned} $$