 |
| a. | $\begin{align}x(x+5) = 0\\[5pt]
\text {x=0 or x+5}=0\\[5pt]
\underline{\underline {\text {x=0 or x=-5}}}\\
\end{align}$
| b. | $\begin{align}\frac {3}{4}x (x+1) = 0\\[5pt]
\frac {3}{4}\text{x=0 or x+1=0}\\[5pt]
\underline{\underline {\text {x=0 or x=-1}}}\\
\end{align}$
|
| c. | $\begin{align}(x-4)(x+3) = 0\\[5pt]
\text {x=-4 or x+3}=0\\[5pt]
\underline{\underline {\text {x=4 or x=-3}}}\\
\end{align}$ | d. | $\begin{align}x^2-2x = 0\\[5pt]
\text {x(x-2)}=0\\[5pt]
\text {x=0 or x-2}=0\\[5pt]
\underline{\underline {\text {x=0 or x=2}}}\\
\end{align}$
|
| e. | $\begin{align}\frac{x^2}{2} = 3x\\[5pt]
x^2=6x\\[5pt]
x^2-6x=0\\[5pt]
x(x-6)=0\\[5pt]
\text {x=0 or x-6}=0\\[5pt]
\underline{\underline {\text {x=0 or x=6}}}\\
\end{align}$
| f. | $\begin{align}x^2+7x+12 = 0\\[5pt]
x^2+3x+4x+12=0\\[5pt]
x(x+3)+4(x+3)=0\\[5pt]
(x+3)(x+4)=0\\[5pt]
\text {x+3=0 or x+4}=0\\[5pt]
\underline{\underline {\text {x=-3 or x=-4}}}\\
\end{align}$
|
| g. | $\begin{align}(x-2)(2x+3) =x^2+2x+4\\[5pt]
2x^2+3x-4x-6=x^2+2x+4\\[5pt]
2x^2-x^2+3x-4x-2x-6-4=0\\[5pt]
x^2-3x-10=0\\[5pt]
(x-5)(x+2)=0\\[5pt]
\text {x-5=0 or x+2}=0\\[5pt]
\underline{\underline {\text {x=5 or x=-2}}}\\
\end{align}$
| h. | $\begin{align}\frac{4}{x}+\frac{3}{x+1}=3\\[5pt]
\frac{4(x+1)+3x}{x(x+1)}=3\\[5pt]
\frac{4x+4+3x}{x^2+x}=3\\[5pt]
7x+4=3x^2+3x\\[5pt]
3x^2-4x-4=0\\[5pt]
(x-2)(3x+2)=0\\[5pt]
\text {x-2=0 or 3x+2}=0\\[5pt]
\underline{\underline {\text {x=2 or } x=-\frac{2}{3}}}\\
\end{align}$
|
| i. | $\begin{align}\frac{2}{x-1}+\frac{3}{x+1}=1\\[5pt]
\frac{2(x+1)+3(x-1)}{(x-1)(x+1)}=1\\[5pt]
\frac{2x+2+3x-3}{x^2-1}=1\\[5pt]
5x-1=x^2-1\\[5pt]
x^2-5x=0\\[5pt]
x(x-5)=0\\[5pt]
\text {x=0 or x-5}=0\\[5pt]
\underline{\underline {\text {x=0 or } x=5}}\\
\end{align}$
| j. | $\begin{align}x^2-4 = 0\\[5pt]
\text {(x+2)(x-2)}=0\\[5pt]
\text {x-2=0 or x+2}=0\\[5pt]
\underline{\underline {\text {x=2 or x=-2}}}\\
\end{align}$
|
 |
| a. | $\begin{align}x^2-12 &= 0\\[5pt]
x^2-(\sqrt {12})^2&=0\\[5pt]
(x-\sqrt {12}) (x+\sqrt {12})&=0\\[5pt]
(x-\sqrt {12})=0 \text { }or \text { } (x+\sqrt {12})=0\\[5pt]
x=\sqrt {12} \text { } or \text { } x=-\sqrt {12}\\[5pt]
x=\sqrt {4\times3} \text { } or \text { } x=-\sqrt {4\times 3}\\[5pt]
x=2\sqrt {3} \text { } or \text { } x=-2\sqrt {3}\\[5pt]
x=2\times 1.73 \text { } or \text { } x=-2\times1.73\\[5pt]
\underline{\underline {\text {x=3.46 or x=-3.46}}}\\
\end{align}$
|
b.
| $\begin{align}x^2-21 &= 11\\[5pt]
x^2-32&=0\\[5pt]
x^2-(\sqrt {32})^2&=0\\[5pt]
(x-\sqrt {32}) (x+\sqrt {32})&=0\\[5pt]
(x-\sqrt {32})=0 \text { }or \text { } (x+\sqrt {32})=0\\[5pt]
x=\sqrt {32} \text { } or \text { } x=-\sqrt {32}\\[5pt]
x=\sqrt {16\times2} \text { } or \text { } x=-\sqrt {16\times 2}\\[5pt]
x=4\sqrt {2} \text { } or \text { } x=-4\sqrt {2}\\[5pt]
x=4\times 1.41 \text { } or \text { } x=-4\times1.41\\[5pt]
\underline{\underline {\text {x=5.64or x=-5.64}}}\\
\end{align}$
|
c.
|
$\begin{align}x^2+17 &= 37\\[5pt]
x^2-20&=0\\[5pt]
x^2-(\sqrt {20})^2&=0\\[5pt]
(x-\sqrt {20}) (x+\sqrt {20})&=0\\[5pt]
(x-\sqrt {20})=0 \text { }or \text { } (x+\sqrt {20})=0\\[5pt]
x=\sqrt {20} \text { } or \text { } x=-\sqrt {20}\\[5pt]
x=\sqrt {4\times5} \text { } or \text { } x=-\sqrt {4\times 5}\\[5pt]
x=2\sqrt {5} \text { } or \text { } x=-2\sqrt {5}\\[5pt]
x=2\times 2.23 \text { } or \text { } x=-2\times 2.23\\[5pt]
\underline{\underline {\text {x=4.46or x=-4.46}}}\\
\end{align}$
| | |
 |
| $\begin{align}x^2-2x &= 15\\[5pt]
x^2-2x-15&=0\\[5pt]
x^2-5x+3x-15&=0\\[5pt]
x(x-5)+3(x-5)&=0\\[5pt]
(x-5)(x+3)&=0\\[5pt]
x-5=0 \text { } or \text { } x+3=0\\[5pt]
\underline{\underline {\text {x=5 or x=-3}}}\\
\end{align}$
| | |
 |
| $\begin{align}x(x+2) &= 120\\[5pt]
x^2+2x-120&=0\\[5pt]
x^2+12x-10x-120&=0\\[5pt]
x(x+12)-10(x+12)&=0\\[5pt]
(x+12)(x-10)&=0\\[5pt]
x+12=0 \text { } or \text { } x-10=0\\[5pt]
\underline{\underline {\text {x=10 , 12 or x=-12 , -10}}}\\
\end{align}$
| | |
 |
| $\begin{align} \text{width=x , length =x+3}\\[5pt]
x(x+3) &= 88\\[5pt]
x^2+3x-88&=0\\[5pt]
x^2+11x-8x-88&=0\\[5pt]
x(x+11)-8(x+11)&=0\\[5pt]
(x+11)(x-8)&=0\\[5pt]
x+11=0 \text { } or \text { } x-8=0\\[5pt]
\underline{\underline {\text {x=-11 or x=8}}}\\
\text{width=8cm , length =11cm}\\[5pt]
\end{align}$
| | |
 |
| $\begin{align} \text{Area of the playing field with road=(32+2x)(20+2x)}\\[5pt]
\text{Area of the playing field =32}\times 20\\[5pt]
\text{Area of the road =(32+2x)(20+2x)-32}\times 20\\[5pt]
(32+2x)(20+2x)-32\times 20&=285\\[5pt]
640+64x+40x+4x^2-640&=285\\[5pt]
4x^2+104x-285&=0\\[5pt]
4x^2+114x-10x-285&=0\\[5pt]
2x(2x+57)-5(2x+57)&=0\\[5pt]
(2x+57)(2x-5)&=0\\[5pt]
2x+57=0 \text { } or \text { } 2x-5=0\\[5pt]
2x=-57 \text { } or \text { } 2x=5\\[5pt]
\underline{\underline {\text {x=-28.5 or x=2.5}}}\\
\text{width of the pathway=2.5m}\\[5pt]
\end{align}$
| | |
 |
 | $\begin{align} AB^2=AC^2+CB^2 \text{(Pythagoras theorem)}\\[5pt]
(2x+1)^2&=(x)^2+(x+7)^2\\[5pt]
4x^2+4x+1&=x^2+x^2+14x+49\\[5pt]
2x^2-10x-48&=0\\[5pt]
x^2-5x-24&=0\\[5pt]
x^2-8x+3x-24&=0\\[5pt]
x(x-8)+3(x-8)&=0\\[5pt]
(x-8)(x+3)&=0\\[5pt]
x-8=0 \text { } or \text { } x+3=0\\[5pt]
x=8 \text { } or \text { } x=-3\\[5pt]
\underline{\underline {\text {x=8}}}\\
\text{AC=8cm}\\[5pt]
\text{CB=x+7=8+7=15cm}\\[5pt]
\text{AB=2x+1=2}\times8+1=17cm\\[5pt]
\end{align}$
| | |
 |
| $\begin{align} (i)\text{-7 , -5 , -3 ,-1 , .....}\\[5pt]
\text{a=-7 , d=2 , Sn=105 , n= ?}\\[5pt]
Sn=\frac{n}{2}\{2a+(n-1)d\}\\[5pt]
105=\frac{n}{2}\{2\times(-7)+(n-1)\times 2\}\\[5pt]
105=\frac{n}{2}\{-14+2n-2\}\\[5pt]
105=\frac{n}{2}\{-16+2n\}\\[5pt]
105=n\{-8+n\}\\[5pt]
105=-8n+n^2\\[5pt]
n^2-8n-105=0\\[5pt]
(ii)n^2-15n+7n-105=0\\[5pt]
n(n-15)+7(n-15)=0\\[5pt]
(n-15)(n+7)=0\\[5pt]
n-15=0 \text { } or \text { } n+7=0\\[5pt]
n=15 \text { } or \text { } n=-7\\[5pt]
\underline{\underline {\text {n=15}}}\\
\end{align}$
| | |