Soalan 3:
Selesaikan setiap persamaan kuadratik berikut.
(a) 4x2 – 1 = 0
(b) x2 – 81 = 0
(c) y2 – 4y = 0
(d) x2+ 3x + 2 = 0
(e) 2x2 – x – 10 = 0
(f) (x – 2)2 = 16
(g) m2 + 3m– 4 = 0
(h) 2p2 – 13p + 20 = 0
(i) (k – 4)(k – 1) = 18
(j) (h – 1)/3 = 1/ (h + 1)
(k) 2(x – 2)2 = 5x – 7
Penyelesaian:
(a)
$$ \begin{gathered} 4 x^2-1=0 \\ (2 x)^2-1^2=0 \\ (2 x-1)(2 x+1)=0 \\ 2 x-1=0 \quad 2 x+1=0 \\ x=\frac{1}{2} \quad x=-\frac{1}{2} \end{gathered} $$
(b)
$$ \begin{aligned} & x^2-81=0 \\ & x^2-9^2=0 \\ & (x-9)(x+9)=0 \\ & x=9 \text { atau } x=-9 \end{aligned} $$
(c)
$$ \begin{aligned} & y^2-4 y=0 \\ & y(y-4)=0 \\ & y=0 \text { atau } y=4 \end{aligned} $$
(d)
$$ \begin{gathered} x^2+3 x+2=0 \\ (x+1)(x+2)=0 \\ x=-1 \quad \text { atau } x=-2 \end{gathered} $$
(e)
$$ \begin{array}{r} 2 x^2-x-10=0 \\ (2 x-5)(x+2)=0 \\ x=\frac{5}{2} \text { atau } x=-2 \end{array} $$
(f)
$$ \begin{aligned} (x-2)^2 & =16 \\ (x-2)(x-2)-16 & =0 \\ x^2-2 x-2 x+4-16 & =0 \\ x^2-4 x-12 & =0 \\ (x-6)(x+2) & =0 \\ x=6 \text { atau } x & =-2 \end{aligned} $$
(g)
$$ \begin{gathered} m^2+3 m-4=0 \\ (m+4)(m-1)=0 \\ m=-4 \text { atau } m=1 \end{gathered} $$
(h)
$$ \begin{aligned} 2 p^2-13 p+20 & =0 \\ (2 p-5)(p-4) & =0 \\ p=\frac{5}{2} \text { atau } p & =4 \end{aligned} $$
(i)
$$ \begin{aligned} (k-4)(k-1) & =18 \\ k^2-k-4 k+4-18 & =0 \\ k^2-5 k-14 & =0 \\ (k-7)(k+2) & =0 \\ k=7 \text { atau } k & =-2 \end{aligned} $$
(j)
$$ \begin{aligned} \frac{h-1}{3} & =\frac{1}{h+1} \\ (h-1)(h+1) & =1 \times 3 \\ h^2+h-h-1 & =3 \\ h^2-1-3 & =0 \\ h^2-4 & =0 \\ h^2-2^2 & =0 \\ (h-2)(h+2) & =0 \\ h=2 \text { atau } h & =-2 \end{aligned} $$
(k)
$$ \begin{aligned} 2(x-2)^2 & =5 x-7 \\ 2(x-2)(x-2) & =5 x-7 \\ 2\left(x^2-2 x-2 x+4\right) & =5 x-7 \\ 2\left(x^2-4 x+4\right) & =5 x-7 \\ 2 x^2-8 x+8-5 x+7 & =0 \\ 2 x^2-13 x+15 & =0 \\ (2 x-3)(x-5) & =0 \\ x=\frac{3}{2} \text { atau } x & =5 \end{aligned} $$
Selesaikan setiap persamaan kuadratik berikut.
(a) 4x2 – 1 = 0
(b) x2 – 81 = 0
(c) y2 – 4y = 0
(d) x2+ 3x + 2 = 0
(e) 2x2 – x – 10 = 0
(f) (x – 2)2 = 16
(g) m2 + 3m– 4 = 0
(h) 2p2 – 13p + 20 = 0
(i) (k – 4)(k – 1) = 18
(j) (h – 1)/3 = 1/ (h + 1)
(k) 2(x – 2)2 = 5x – 7
Penyelesaian:
(a)
$$ \begin{gathered} 4 x^2-1=0 \\ (2 x)^2-1^2=0 \\ (2 x-1)(2 x+1)=0 \\ 2 x-1=0 \quad 2 x+1=0 \\ x=\frac{1}{2} \quad x=-\frac{1}{2} \end{gathered} $$
(b)
$$ \begin{aligned} & x^2-81=0 \\ & x^2-9^2=0 \\ & (x-9)(x+9)=0 \\ & x=9 \text { atau } x=-9 \end{aligned} $$
(c)
$$ \begin{aligned} & y^2-4 y=0 \\ & y(y-4)=0 \\ & y=0 \text { atau } y=4 \end{aligned} $$
(d)
$$ \begin{gathered} x^2+3 x+2=0 \\ (x+1)(x+2)=0 \\ x=-1 \quad \text { atau } x=-2 \end{gathered} $$
(e)
$$ \begin{array}{r} 2 x^2-x-10=0 \\ (2 x-5)(x+2)=0 \\ x=\frac{5}{2} \text { atau } x=-2 \end{array} $$
(f)$$ \begin{aligned} (x-2)^2 & =16 \\ (x-2)(x-2)-16 & =0 \\ x^2-2 x-2 x+4-16 & =0 \\ x^2-4 x-12 & =0 \\ (x-6)(x+2) & =0 \\ x=6 \text { atau } x & =-2 \end{aligned} $$
(g)
$$ \begin{gathered} m^2+3 m-4=0 \\ (m+4)(m-1)=0 \\ m=-4 \text { atau } m=1 \end{gathered} $$
(h)
$$ \begin{aligned} 2 p^2-13 p+20 & =0 \\ (2 p-5)(p-4) & =0 \\ p=\frac{5}{2} \text { atau } p & =4 \end{aligned} $$
(i)
$$ \begin{aligned} (k-4)(k-1) & =18 \\ k^2-k-4 k+4-18 & =0 \\ k^2-5 k-14 & =0 \\ (k-7)(k+2) & =0 \\ k=7 \text { atau } k & =-2 \end{aligned} $$
(j)
$$ \begin{aligned} \frac{h-1}{3} & =\frac{1}{h+1} \\ (h-1)(h+1) & =1 \times 3 \\ h^2+h-h-1 & =3 \\ h^2-1-3 & =0 \\ h^2-4 & =0 \\ h^2-2^2 & =0 \\ (h-2)(h+2) & =0 \\ h=2 \text { atau } h & =-2 \end{aligned} $$
(k)
$$ \begin{aligned} 2(x-2)^2 & =5 x-7 \\ 2(x-2)(x-2) & =5 x-7 \\ 2\left(x^2-2 x-2 x+4\right) & =5 x-7 \\ 2\left(x^2-4 x+4\right) & =5 x-7 \\ 2 x^2-8 x+8-5 x+7 & =0 \\ 2 x^2-13 x+15 & =0 \\ (2 x-3)(x-5) & =0 \\ x=\frac{3}{2} \text { atau } x & =5 \end{aligned} $$
Soalan 4:
Diberi salah satu punca bagi persamaan kuadratik x2 + px – 18 = 0 ialah 2. Hitung nilai p.
Penyelesaian:
$$ \begin{aligned} &\text { Diberi salah satu punca, } x=2\\ &\begin{aligned} x^2+p x-18 & =0 \\ (2)^2+p(2)-18 & =0 \\ 4+2 p-18 & =0 \\ 2 p-14 & =0 \\ 2 p & =14 \\ p & =7 \end{aligned} \end{aligned} $$
Diberi salah satu punca bagi persamaan kuadratik x2 + px – 18 = 0 ialah 2. Hitung nilai p.
Penyelesaian:
$$ \begin{aligned} &\text { Diberi salah satu punca, } x=2\\ &\begin{aligned} x^2+p x-18 & =0 \\ (2)^2+p(2)-18 & =0 \\ 4+2 p-18 & =0 \\ 2 p-14 & =0 \\ 2 p & =14 \\ p & =7 \end{aligned} \end{aligned} $$