Question
Solve : $8 \times 2^{2 x}+4 \times 2^{x+1}=1+2^x$
✓
Answer
$8 \times 2^{2 x}+4 \times 2^{x+1}=1+2^x$
$ \Rightarrow 8 \times\left(2^x\right)^2+4 \times 2^x \times 2^1=1+2^x$
$ \Rightarrow 8 \times\left(2^x\right)^2+4 \times 2^x \times 2^1-1-2^x=0$
$ \Rightarrow 8 \times\left(2^x\right)^2+2^x \times(8-1)-1=0$
$ \Rightarrow 8 \times\left(2^x\right)^2+7\left(2^x\right)-1=0$
$\Rightarrow 8 y^2+7 y-1=0$
$\left[y=2^x\right]$
$ \Rightarrow 8 y^2+8 y-y-1=0$
$ \Rightarrow 8 y(y+1)-1(y+1)=0$
$ \Rightarrow(8 y-1)(y+1)=0$
$ \Rightarrow 8 y=1$ or $y=-1$
$ \Rightarrow y=\frac{1}{8}$ or $y=-1$
$ \Rightarrow 2^x=\frac{1}{8}$ or $2^x=-1$
$ \Rightarrow 2^x=\frac{1}{2^3}$ or $2^x=-1$
$ \Rightarrow 2^x=2^{-3}$ or $2^x=-1$
$ \Rightarrow x=-3$
$[\because 2^x=-1$ is not possible$]$
$ \Rightarrow 8 \times\left(2^x\right)^2+4 \times 2^x \times 2^1=1+2^x$
$ \Rightarrow 8 \times\left(2^x\right)^2+4 \times 2^x \times 2^1-1-2^x=0$
$ \Rightarrow 8 \times\left(2^x\right)^2+2^x \times(8-1)-1=0$
$ \Rightarrow 8 \times\left(2^x\right)^2+7\left(2^x\right)-1=0$
$\Rightarrow 8 y^2+7 y-1=0$
$\left[y=2^x\right]$
$ \Rightarrow 8 y^2+8 y-y-1=0$
$ \Rightarrow 8 y(y+1)-1(y+1)=0$
$ \Rightarrow(8 y-1)(y+1)=0$
$ \Rightarrow 8 y=1$ or $y=-1$
$ \Rightarrow y=\frac{1}{8}$ or $y=-1$
$ \Rightarrow 2^x=\frac{1}{8}$ or $2^x=-1$
$ \Rightarrow 2^x=\frac{1}{2^3}$ or $2^x=-1$
$ \Rightarrow 2^x=2^{-3}$ or $2^x=-1$
$ \Rightarrow x=-3$
$[\because 2^x=-1$ is not possible$]$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The cost of a machine depreciated by $Rs. 4,000$ during the first year and by $Rs. 3,600$ during the second year. Calculate :
→- The rate of depreciation.
- The original cost of the machine.
- Its cost at the end of the third year.
The class $XI$ students of school wanted to give a farewell party to the outgoing students of class $XII$. They decided to purchase two kinds of sweets, one costing $Rs. 250$ per $kg$ and other costing $Rs. 350$ per $kg$. They estimated that $40 \ kg$ of sweets were needed. If the total budget for the sweets was $Rs. 11,800$; find how much sweets of each kind were bought ?
→If $10y = 7x - 4$ and $12x + 18y = 1 ;$ find the value of $4x + 6y$ and $8y - x.$
→Make $f$ the subject of the formula $D=\sqrt{\left(\frac{f+p}{f-p}\right)}$. Find $f_r$ when $D=13$ and $P=21$.
→In the given figure, ABCD is a trapezium in which $AD =13 cm, BC =5 cm, CD =17 cm$ and $\angle A =\angle B =90$.
Calculate :
(i) AB
(ii) Area of trap. ABCD .
→Calculate :
(i) AB
(ii) Area of trap. ABCD .
Insert three rational number between:$-3$ and $3$
→$A$ and $B$ together can complete a piece of work in $6$ days. $A$ can do it alone in $10$ days. Find the number of days in which $B$ alone can do the work.
→Find the interior angles of the following triangles:
Find the amount and the compound interest payable annually on:$Rs.16000$ for $2$ years at $15\%$ and $12\%$ for the successive years.
→Construct a regular hexagon of side $3.2 \ cm$
→